[math-fun] A grid with MAX
Hello MathFunsters, I've seen this on "Le Monde" webpage. http://bit.ly/12RaSEi Draw a 3x3 square and write "1" in the upper left corner-square. Fill now one by one the other squares according to these rules: - select an empty square - write in it the sum of its neighboring squares (a "corner-square" has 3 neighbours, an "edge-square" 5 and the "center-square" 8) - when all squares are marked, record the highest written value. Example: +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | 22 | 2 | 6 | <--- MAX value = 22 | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ My best MAX is 40: is it the highest possible MAX in a 3x3 square? What are, if we consider all nxn squares, the MAX values for the first n? The sequence of such MAX starts with S(max) = 1,4,... then what? This is perhaps old hat, sorry -- but there are too many seqs in the OEIS that start with 1,4,... Best, E.
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44. If we label the 9 squares 1-9: +-----+-----+-----+ | 1 | 2 | 3 | | | | | +-----+-----+-----+ | 4 | 5 | 6 | | | | | +-----+-----+-----+ | 7 | 8 | 9 | | | | | +-----+-----+-----+ then I get 44 by doing the squares in the order: {2, 4, 7, 5, 3, 6, 8, 9} {2, 4, 7, 5, 3, 6, 9, 8} or their symmetrical counterparts {4, 2, 3, 5, 7, 8, 6, 9} {4, 2, 3, 5, 7, 8, 9, 6}
Thanks Hans, Erich Friedman confirms (in a private mail -- his posts are rejected by MFun) your 44. The sequence of nxn MAX is perhaps somewhere here: http://oeis.org/search?q=1%2C4%2C44&language=english&go=Search Best, É. Propulsé d'un aPhone Le 14 avr. 2013 à 20:02, "Hans Havermann" <gladhobo@teksavvy.com> a écrit :
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44. If we label the 9 squares 1-9:
+-----+-----+-----+ | 1 | 2 | 3 | | | | | +-----+-----+-----+ | 4 | 5 | 6 | | | | | +-----+-----+-----+ | 7 | 8 | 9 | | | | | +-----+-----+-----+
then I get 44 by doing the squares in the order:
{2, 4, 7, 5, 3, 6, 8, 9} {2, 4, 7, 5, 3, 6, 9, 8}
or their symmetrical counterparts
{4, 2, 3, 5, 7, 8, 6, 9} {4, 2, 3, 5, 7, 8, 9, 6}
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 4/14/2013 2:00 PM, Hans Havermann wrote:
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44.
I concur with Hans, and add that the highest for a 4x4 square seems to be 2473 (barring software or hardware errors). The filled-in grid looks like this: 1 1 1239 2473 2 4 419 815 6 18 100 296 6 30 48 148 It's easy to figure out the order in which the numbers are entered. The two 6s are a bit surprising. My brute force approach handles 3x3 in under a second and 4x4 in under two hours; 5x5 would seem to be infeasible without better ideas. -- Fred W. Helenius fredh@ix.netcom.com
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably. On Sun, Apr 14, 2013 at 4:14 PM, Fred W. Helenius <fredh@ix.netcom.com>wrote:
On 4/14/2013 2:00 PM, Hans Havermann wrote:
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44.
I concur with Hans, and add that the highest for a 4x4 square seems to be 2473 (barring software or hardware errors). The filled-in grid looks like this:
1 1 1239 2473
2 4 419 815
6 18 100 296
6 30 48 148
It's easy to figure out the order in which the numbers are entered. The two 6s are a bit surprising.
My brute force approach handles 3x3 in under a second and 4x4 in under two hours; 5x5 would seem to be infeasible without better ideas.
-- Fred W. Helenius fredh@ix.netcom.com
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid: 1 1 53181 106357 143018 2 4 16515 36661 297136 6 12 9046 7452 3648 18 54 156 1368 2280 18 90 300 456 456 Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks. As ever, independent confirmation would be welcome. -- Fred W. Helenius fredh@ix.netcom.com
Haha, "helpy" Gmail hyperlinked the three numbers 156 1368 2280, because it looked just enough like a phone number. My intuition has crystallized just a little bit. If you have a path that is not "contiguous", that means there is a pair of cells c[n], c[n+1] that are not kingwise adjacent. Sometime after c[n+1] there occurs the first cell c[n+k] that *is* adjacent to c[n]. If you alter the cell sequence by moving c[n+k] up to follow c[n] immediately, shoving everything else down one space to make room -- then how can that hurt? It seems ("seems" -- this is not a proof!) that the last element can't decrease. I just now got a little bit less confident, but maybe somebody else can either push this through or find a counterexample. On Sun, Apr 14, 2013 at 7:29 PM, Fred W. Helenius <fredh@ix.netcom.com>wrote:
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid:
1 1 53181 106357 143018
2 4 16515 36661 297136
6 12 9046 7452 3648
18 54 156 1368 2280
18 90 300 456 456
Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks.
As ever, independent confirmation would be welcome.
-- Fred W. Helenius fredh@ix.netcom.com
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder. Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of H(n) = 1 + T(n-1) hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n= 0,1,2,.... (Of course for the hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge.) --Dan On 2013-04-14, at 4:42 PM, Allan Wechsler wrote:
Haha, "helpy" Gmail hyperlinked the three numbers 156 1368 2280, because it looked just enough like a phone number.
My intuition has crystallized just a little bit. If you have a path that is not "contiguous", that means there is a pair of cells c[n], c[n+1] that are not kingwise adjacent. Sometime after c[n+1] there occurs the first cell c[n+k] that *is* adjacent to c[n]. If you alter the cell sequence by moving c[n+k] up to follow c[n] immediately, shoving everything else down one space to make room -- then how can that hurt? It seems ("seems" -- this is not a proof!) that the last element can't decrease. I just now got a little bit less confident, but maybe somebody else can either push this through or find a counterexample.
On Sun, Apr 14, 2013 at 7:29 PM, Fred W. Helenius <fredh@ix.netcom.com>wrote:
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid:
1 1 53181 106357 143018
2 4 16515 36661 297136
6 12 9046 7452 3648
18 54 156 1368 2280
18 90 300 456 456
Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks.
As ever, independent confirmation would be welcome.
-- Fred W. Helenius fredh@ix.netcom.com
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I was contemplating another generalization, this one to mxn grids. 2xn is interesting, because the optimal paths on square grids always seem to start by sweeping out a 2xn rectangle first. On Sun, Apr 14, 2013 at 8:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder.
Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of
H(n) = 1 + T(n-1)
hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n= 0,1,2,....
(Of course for the hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge.)
--Dan
On 2013-04-14, at 4:42 PM, Allan Wechsler wrote:
Haha, "helpy" Gmail hyperlinked the three numbers 156 1368 2280, because it looked just enough like a phone number.
My intuition has crystallized just a little bit. If you have a path that is not "contiguous", that means there is a pair of cells c[n], c[n+1] that are not kingwise adjacent. Sometime after c[n+1] there occurs the first cell c[n+k] that *is* adjacent to c[n]. If you alter the cell sequence by moving c[n+k] up to follow c[n] immediately, shoving everything else down one space to make room -- then how can that hurt? It seems ("seems" -- this is not a proof!) that the last element can't decrease. I just now got a little bit less confident, but maybe somebody else can either push this through or find a counterexample.
On Sun, Apr 14, 2013 at 7:29 PM, Fred W. Helenius <fredh@ix.netcom.com wrote:
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid:
1 1 53181 106357 143018
2 4 16515 36661 297136
6 12 9046 7452 3648
18 54 156 1368 2280
18 90 300 456 456
Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks.
As ever, independent confirmation would be welcome.
-- Fred W. Helenius fredh@ix.netcom.com
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<; http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Correction to my last post: I misstated the size of the nth "hex number" H(n). This is correct (rest of post repeated with annotation for context): ----- As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder. Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of H(n) = 1 + 6*T(n-1) hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n=0,1,2,.... The sequence of H(n) is 1, 7, 19, 37, 61, 91, 127,.... But in fact, there are many other-sized hexagonal groups of hexagons that can have opposite sides identified consistently to make a hexagonal torus. The most natural of these is to iterate making a rosette of the previous pre-identification configuration, giving the sequence of powers 7^n of 7 as the cell counts: 1, 7, 49, 343, 2401,.... (Of course for any hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge. ----- --Dan On 2013-04-14, at 5:19 PM, Allan Wechsler wrote:
As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder.
Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of
H(n) = 1 + T(n-1)
hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n= 0,1,2,....
(Of course for the hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge.)
Allan Wechsler:
I was contemplating another generalization, this one to mxn grids.
I've got the 3x4 as: 1 1 138 269 2 6 33 98 2 10 16 49 The second solution has 269, 131 at the top-right. It appears to be true, for any grid that I've looked at, that order does not matter for the final two positions, even though the second-last number may (will?) be different in the two cases. I can't do an exhaustive 3x5 (I'm sure Fred will verify or refute) but here is a candidate: 1 1 40 73 235 2 6 33 162 470 2 10 16 681 1313
Dan, I believe that the point were symmetry in a classical sense has the most influence (or the clearest) is when playing for worst score instead of best. For instance, on the 3x3 square grid, the "smallest" greatest cell you can have is 8 I believe. But on a 3x3 torus, every cell is the neighbour of every other so all paths are equivalent and you end up with 128 = 2^7. etc. The sequence corresponding to the worst score in a nxn grid for strict rules seems to be 1,4, 8, 32 ?, 28 ? , ... Olivier On Mon, Apr 15, 2013 at 2:16 AM, Dan Asimov <dasimov@earthlink.net> wrote:
As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder.
Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of
H(n) = 1 + T(n-1)
hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n= 0,1,2,....
(Of course for the hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge.)
--Dan
Yes, 297136 seems to be the best score for 5x5. Interesting and "math-fun", there are 3 other solutions with the same score, using different paths and different intermediate numbers. One of them: 1 1 40 73 235 2 6 33 162 470 2 10 16 1313 632 148574 49530 16947 3906 1945 297136 99032 32555 11702 5851 If we sum the 25 numbers of this solution, we obtain 670174. Sums of the four solutions scoring 297136 are: 670162, 670174 (above), 678236 (Fred), 689336. The last one is close to Fred's, only inverting the two last moves: 1 1 53181 106357 297136 2 4 16515 36661 154118 6 12 9046 7452 3648 18 54 156 1368 2280 18 90 300 456 456 Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Fred W. Helenius Envoyé : lundi 15 avril 2013 01:29 À : math-fun@mailman.xmission.com Objet : Re: [math-fun] A grid with MAX On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid: 1 1 53181 106357 143018 2 4 16515 36661 297136 6 12 9046 7452 3648 18 54 156 1368 2280 18 90 300 456 456 Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks. As ever, independent confirmation would be welcome. -- Fred W. Helenius fredh@ix.netcom.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Are all paths (for max) connected? Is there always an optimal path that is greedy (chooses at every step one of the squares that has the highest value from that point)? Searching greedy sequences only would seem to be reasonably fast and productive . . . On Mon, Apr 15, 2013 at 11:26 AM, Christian Boyer <cboyer@club-internet.fr> wrote:
Yes, 297136 seems to be the best score for 5x5. Interesting and "math-fun", there are 3 other solutions with the same score, using different paths and different intermediate numbers. One of them:
1 1 40 73 235 2 6 33 162 470 2 10 16 1313 632 148574 49530 16947 3906 1945 297136 99032 32555 11702 5851
If we sum the 25 numbers of this solution, we obtain 670174.
Sums of the four solutions scoring 297136 are: 670162, 670174 (above), 678236 (Fred), 689336. The last one is close to Fred's, only inverting the two last moves:
1 1 53181 106357 297136 2 4 16515 36661 154118 6 12 9046 7452 3648 18 54 156 1368 2280 18 90 300 456 456
Christian.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Fred W. Helenius Envoyé : lundi 15 avril 2013 01:29 À : math-fun@mailman.xmission.com Objet : Re: [math-fun] A grid with MAX
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid:
1 1 53181 106357 143018
2 4 16515 36661 297136
6 12 9046 7452 3648
18 54 156 1368 2280
18 90 300 456 456
Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks.
As ever, independent confirmation would be welcome.
-- Fred W. Helenius fredh@ix.netcom.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- http://golly.sf.net/ --
Oops; never mind. The 4x4 case has no greedy and optimal solution. On Mon, Apr 15, 2013 at 11:34 AM, Tom Rokicki <rokicki@gmail.com> wrote:
Are all paths (for max) connected?
Is there always an optimal path that is greedy (chooses at every step one of the squares that has the highest value from that point)?
Searching greedy sequences only would seem to be reasonably fast and productive . . .
On Mon, Apr 15, 2013 at 11:26 AM, Christian Boyer <cboyer@club-internet.fr> wrote:
Yes, 297136 seems to be the best score for 5x5. Interesting and "math-fun", there are 3 other solutions with the same score, using different paths and different intermediate numbers. One of them:
1 1 40 73 235 2 6 33 162 470 2 10 16 1313 632 148574 49530 16947 3906 1945 297136 99032 32555 11702 5851
If we sum the 25 numbers of this solution, we obtain 670174.
Sums of the four solutions scoring 297136 are: 670162, 670174 (above), 678236 (Fred), 689336. The last one is close to Fred's, only inverting the two last moves:
1 1 53181 106357 297136 2 4 16515 36661 154118 6 12 9046 7452 3648 18 54 156 1368 2280 18 90 300 456 456
Christian.
-----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Fred W. Helenius Envoyé : lundi 15 avril 2013 01:29 À : math-fun@mailman.xmission.com Objet : Re: [math-fun] A grid with MAX
On 4/14/2013 5:43 PM, Allan Wechsler wrote:
Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably.
It's plausible--if only because the 3x3 and 4x4 solutions work that way--but I don't see a proof. But it would be a great thing to know because assuming the correctness of Allan's conjecture reduces the computation for 5x5 to under 10 minutes. The conjectural champion value is then 297136, from this grid:
1 1 53181 106357 143018
2 4 16515 36661 297136
6 12 9046 7452 3648
18 54 156 1368 2280
18 90 300 456 456
Even with this improvement, 6x6 still looks difficult; possibly on the order of CPU-weeks.
As ever, independent confirmation would be welcome.
-- Fred W. Helenius fredh@ix.netcom.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- http://golly.sf.net/ --
-- -- http://cube20.org/ -- http://golly.sf.net/ --
90134672 is my best score on 6x6: 1 2 2 90134672 32354320 16177160 1 6 10 46664413 11115927 5061233 40 33 16 993461 2200677 2860556 73 162 211 547337 445887 213992 681 446 3073 17903 80247 133745 1127 2254 5773 8846 26749 26749 Far to be a result of an exhaustive search! Maybe possible to improve? Christian.
;-!!!! Bravo Christian, pas mieux ! É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : mardi 16 avril 2013 12:42 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX 90134672 is my best score on 6x6: 1 2 2 90134672 32354320 16177160 1 6 10 46664413 11115927 5061233 40 33 16 993461 2200677 2860556 73 162 211 547337 445887 213992 681 446 3073 17903 80247 133745 1127 2254 5773 8846 26749 26749 Far to be a result of an exhaustive search! Maybe possible to improve? Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Christian Boyer:
90134672 is my best score on 6x6:
I managed a 102568856 on my first manual try and 108885920 on my second, which may include, of course, arithmetical errors. I'm sure we're going to see better numbers. Because the manual arithmetic is such a pain, this would make a fun mobile app. On a reasonably large board (not necessarily square) you just keep tapping the next square in an attempt to beat your previous best.
Right, we will see better numbers. My current best result for 6x6 is 128493464. Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Hans Havermann Envoyé : mardi 16 avril 2013 17:28 À : math-fun Objet : Re: [math-fun] A grid with MAX Christian Boyer:
90134672 is my best score on 6x6:
I managed a 102568856 on my first manual try and 108885920 on my second, which may include, of course, arithmetical errors. I'm sure we're going to see better numbers. Because the manual arithmetic is such a pain, this would make a fun mobile app. On a reasonably large board (not necessarily square) you just keep tapping the next square in an attempt to beat your previous best. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Slightly better, 128493518, with an interesting path: 1 2 6 18 54 54 1 4 12 36 162 270 7693637 15387286 42350640 128493518 468 900 2495232 5198400 21764902 64374562 2736 1368 1559520 935712 207936 35568 12312 4104 311904 311904 103968 68400 20520 4104 And repeating numbers in each corner (1, 54, 4104, 311904) Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : mardi 16 avril 2013 18:44 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Right, we will see better numbers. My current best result for 6x6 is 128493464. Christian.
Waow, impressive result, Fred -- many thanks! And to Hans also (private mail). I've just checked: S(max) = 1,4,44,2473,... is not in the OEIS. The solution for n=5 might arise from a search of what could be the best path -- regardless of any computations, having perhaps only patterns in mind (patterns of pathes)... My two (naïve) cents. Best, É. Le 14 avr. 2013 à 22:15, "Fred W. Helenius" <fredh@ix.netcom.com> a écrit :
On 4/14/2013 2:00 PM, Hans Havermann wrote:
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44.
I concur with Hans, and add that the highest for a 4x4 square seems to be 2473 (barring software or hardware errors). The filled-in grid looks like this:
1 1 1239 2473
2 4 419 815
6 18 100 296
6 30 48 148
It's easy to figure out the order in which the numbers are entered. The two 6s are a bit surprising.
My brute force approach handles 3x3 in under a second and 4x4 in under two hours; 5x5 would seem to be infeasible without better ideas.
-- Fred W. Helenius fredh@ix.netcom.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hmmm... if we carefully listen (in French) the original 3x3 problem in "Le Monde" website, http://bit.ly/12RaSEi, the rules are different! Eric, maybe you no more speak French? ;-) The BEGINNING of the game is different: we first have to choose two cells (not necessarily neighboring), placing 1 in these two cells. Then the rules are the same: each new chosen cell is filled with the sum of the neighboring cells. With these rules, the new max for 3x3 is no more 44, but 57 (I think): 1 2 1 3 7 30 10 20 57 However, this should not change the records of bigger grids. On 4x4, 5x5,... it's more interesting to place the two first 1 in the same corner, meaning that the rules become equivalent. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : dimanche 14 avril 2013 16:12 À : math-fun Cc : alexandre.wajnberg@skynet.be Objet : [math-fun] A grid with MAX Hello MathFunsters, I've seen this on "Le Monde" webpage. http://bit.ly/12RaSEi Draw a 3x3 square and write "1" in the upper left corner-square. Fill now one by one the other squares according to these rules: - select an empty square - write in it the sum of its neighboring squares (a "corner-square" has 3 neighbours, an "edge-square" 5 and the "center-square" 8) - when all squares are marked, record the highest written value. Example: +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | 22 | 2 | 6 | <--- MAX value = 22 | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ My best MAX is 40: is it the highest possible MAX in a 3x3 square? What are, if we consider all nxn squares, the MAX values for the first n? The sequence of such MAX starts with S(max) = 1,4,... then what? This is perhaps old hat, sorry -- but there are too many seqs in the OEIS that start with 1,4,... Best, E. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
the rules are different!
Yes, I know, Christian, Oui, je sais, Christian, But the rules in Le Monde's video Mais les règles dans la vidéo du Monde Are not consistent: Ne sont pas cohérentes : Why does he put a second "1" in an isolated cell with no neighbors? Pourquoi inscrit-il un second "1" dans une cellule sans voisins ? I think that the corrected version is simpler and more logical: Occam's razor! Comprends-je toujours le français ? Ma version est meilleure, non ? Regards, Regards. ;-D É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 18 avril 2013 11:58 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Hmmm... if we carefully listen (in French) the original 3x3 problem in "Le Monde" website, http://bit.ly/12RaSEi, the rules are different! Eric, maybe you no more speak French? ;-) The BEGINNING of the game is different: we first have to choose two cells (not necessarily neighboring), placing 1 in these two cells. Then the rules are the same: each new chosen cell is filled with the sum of the neighboring cells. With these rules, the new max for 3x3 is no more 44, but 57 (I think): 1 2 1 3 7 30 10 20 57 However, this should not change the records of bigger grids. On 4x4, 5x5,... it's more interesting to place the two first 1 in the same corner, meaning that the rules become equivalent. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : dimanche 14 avril 2013 16:12 À : math-fun Cc : alexandre.wajnberg@skynet.be Objet : [math-fun] A grid with MAX Hello MathFunsters, I've seen this on "Le Monde" webpage. http://bit.ly/12RaSEi Draw a 3x3 square and write "1" in the upper left corner-square. Fill now one by one the other squares according to these rules: - select an empty square - write in it the sum of its neighboring squares (a "corner-square" has 3 neighbours, an "edge-square" 5 and the "center-square" 8) - when all squares are marked, record the highest written value. Example: +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | 22 | 2 | 6 | <--- MAX value = 22 | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ My best MAX is 40: is it the highest possible MAX in a 3x3 square? What are, if we consider all nxn squares, the MAX values for the first n? The sequence of such MAX starts with S(max) = 1,4,... then what? This is perhaps old hat, sorry -- but there are too many seqs in the OEIS that start with 1,4,... Best, E. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Sure Eric, your version is much more logical, I like it! And with the two versions, records should be the same for 4x4 and above. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : jeudi 18 avril 2013 12:58 À : math-fun Objet : Re: [math-fun] A grid with MAX
the rules are different!
Yes, I know, Christian, Oui, je sais, Christian, But the rules in Le Monde's video Mais les règles dans la vidéo du Monde Are not consistent: Ne sont pas cohérentes : Why does he put a second "1" in an isolated cell with no neighbors? Pourquoi inscrit-il un second "1" dans une cellule sans voisins ? I think that the corrected version is simpler and more logical: Occam's razor! Comprends-je toujours le français ? Ma version est meilleure, non ? Regards, Regards. ;-D É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 18 avril 2013 11:58 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Hmmm... if we carefully listen (in French) the original 3x3 problem in "Le Monde" website, http://bit.ly/12RaSEi, the rules are different! Eric, maybe you no more speak French? ;-) The BEGINNING of the game is different: we first have to choose two cells (not necessarily neighboring), placing 1 in these two cells. Then the rules are the same: each new chosen cell is filled with the sum of the neighboring cells. With these rules, the new max for 3x3 is no more 44, but 57 (I think): 1 2 1 3 7 30 10 20 57 However, this should not change the records of bigger grids. On 4x4, 5x5,... it's more interesting to place the two first 1 in the same corner, meaning that the rules become equivalent. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : dimanche 14 avril 2013 16:12 À : math-fun Cc : alexandre.wajnberg@skynet.be Objet : [math-fun] A grid with MAX Hello MathFunsters, I've seen this on "Le Monde" webpage. http://bit.ly/12RaSEi Draw a 3x3 square and write "1" in the upper left corner-square. Fill now one by one the other squares according to these rules: - select an empty square - write in it the sum of its neighboring squares (a "corner-square" has 3 neighbours, an "edge-square" 5 and the "center-square" 8) - when all squares are marked, record the highest written value. Example: +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | 22 | 2 | 6 | <--- MAX value = 22 | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ My best MAX is 40: is it the highest possible MAX in a 3x3 square? What are, if we consider all nxn squares, the MAX values for the first n? The sequence of such MAX starts with S(max) = 1,4,... then what? This is perhaps old hat, sorry -- but there are too many seqs in the OEIS that start with 1,4,... Best, E. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
with the two versions, records should be the same for 4x4 and above.
... Yes, you're right ! Best, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 18 avril 2013 13:55 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Sure Eric, your version is much more logical, I like it! And with the two versions, records should be the same for 4x4 and above. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : jeudi 18 avril 2013 12:58 À : math-fun Objet : Re: [math-fun] A grid with MAX
the rules are different!
Yes, I know, Christian, Oui, je sais, Christian, But the rules in Le Monde's video Mais les règles dans la vidéo du Monde Are not consistent: Ne sont pas cohérentes : Why does he put a second "1" in an isolated cell with no neighbors? Pourquoi inscrit-il un second "1" dans une cellule sans voisins ? I think that the corrected version is simpler and more logical: Occam's razor! Comprends-je toujours le français ? Ma version est meilleure, non ? Regards, Regards. ;-D É. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 18 avril 2013 11:58 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Hmmm... if we carefully listen (in French) the original 3x3 problem in "Le Monde" website, http://bit.ly/12RaSEi, the rules are different! Eric, maybe you no more speak French? ;-) The BEGINNING of the game is different: we first have to choose two cells (not necessarily neighboring), placing 1 in these two cells. Then the rules are the same: each new chosen cell is filled with the sum of the neighboring cells. With these rules, the new max for 3x3 is no more 44, but 57 (I think): 1 2 1 3 7 30 10 20 57 However, this should not change the records of bigger grids. On 4x4, 5x5,... it's more interesting to place the two first 1 in the same corner, meaning that the rules become equivalent. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Eric Angelini Envoyé : dimanche 14 avril 2013 16:12 À : math-fun Cc : alexandre.wajnberg@skynet.be Objet : [math-fun] A grid with MAX Hello MathFunsters, I've seen this on "Le Monde" webpage. http://bit.ly/12RaSEi Draw a 3x3 square and write "1" in the upper left corner-square. Fill now one by one the other squares according to these rules: - select an empty square - write in it the sum of its neighboring squares (a "corner-square" has 3 neighbours, an "edge-square" 5 and the "center-square" 8) - when all squares are marked, record the highest written value. Example: +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | | 2 | 6 | | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ +-----+-----+-----+ | 1 | 1 | 3 | | | | | +-----+-----+-----+ | 22 | 2 | 6 | <--- MAX value = 22 | | | | +-----+-----+-----+ | 10 | 8 | 0 | | | | | +-----+-----+-----+ My best MAX is 40: is it the highest possible MAX in a 3x3 square? What are, if we consider all nxn squares, the MAX values for the first n? The sequence of such MAX starts with S(max) = 1,4,... then what? This is perhaps old hat, sorry -- but there are too many seqs in the OEIS that start with 1,4,... Best, E. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Back to Le Monde's version starting with two 1 to be placed where we want. Well... surprise... records are probably NOT the same than the version starting with only one 1. Except only for 4x4 (and the obvious 2x2). I was too fast saying that the rules were equivalent... Not intuitive, but it is often interesting to scatter the two 1! I obtain these better scores, only modifying the beginning of the grids, then using exactly my same "snail" strategy. Results obtained by hand and Excel, not sure that they are the best possible. Only 5X5 is, if no error, the best possible result. ----5x5 score 349712 (vs 297136 with the other version) 1 7 11 33 33 2 4 22 99 165 1 30272 16581 286 550 97477 67204 13662 2508 836 349712 185031 6688 4180 836 ----6x6 score 162857852 (vs 128493518 in the other version) 1 7 11 33 99 99 2 4 22 66 330 528 1 15218576 53335420 162857852 924 1782 4935745 10282802 27833950 81681728 5412 2706 3084840 1850904 411312 70356 24354 8118 616968 616968 205656 135300 40590 8118 ----7x7 score 164912612349 (vs 139015458134 in the other version) ----8x8 score 639561837115046 (vs 541048181546137 in the other version) I can also send these 7x7 and 8x8 grids if somebody wishes, but easy to create using my previous 7x7 and 8x8. http://oeis.org/A221866 need some changes, should be: 1, 4, 57, 2473, 349712 (however, warning, 1x1 can't be played, because we need to place two 1) http://oeis.org/A224784 remain the same: 1, 4, 44, 2473, 297136 Christian.
Oooops, a mistake in my 6x6 grid, and I hope only in this one. The cell with 330 was wrong, this cell should be 33 + 99 + 99 + 66 = 297. An eraser... I refill the grid with the same method... and the correct grid should be: ----6x6 score 150941453 1 7 11 33 99 99 2 4 22 66 297 495 1 14105024 49432840 150941453 858 1650 4574593 9530402 25797322 75705032 5016 2508 2859120 1715472 381216 65208 22572 7524 571824 571824 190608 125400 37620 7524 Hoping that all is OK now. Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 25 avril 2013 16:44 À : 'math-fun' Objet : Re: [math-fun] A grid with MAX Back to Le Monde's version starting with two 1 to be placed where we want. Well... surprise... records are probably NOT the same than the version starting with only one 1. Except only for 4x4 (and the obvious 2x2). I was too fast saying that the rules were equivalent... Not intuitive, but it is often interesting to scatter the two 1! I obtain these better scores, only modifying the beginning of the grids, then using exactly my same "snail" strategy. Results obtained by hand and Excel, not sure that they are the best possible. Only 5X5 is, if no error, the best possible result. ----5x5 score 349712 (vs 297136 with the other version) 1 7 11 33 33 2 4 22 99 165 1 30272 16581 286 550 97477 67204 13662 2508 836 349712 185031 6688 4180 836 ----6x6 score 162857852 (vs 128493518 in the other version) 1 7 11 33 99 99 2 4 22 66 330 528 1 15218576 53335420 162857852 924 1782 4935745 10282802 27833950 81681728 5412 2706 3084840 1850904 411312 70356 24354 8118 616968 616968 205656 135300 40590 8118 ----7x7 score 164912612349 (vs 139015458134 in the other version) ----8x8 score 639561837115046 (vs 541048181546137 in the other version) I can also send these 7x7 and 8x8 grids if somebody wishes, but easy to create using my previous 7x7 and 8x8. http://oeis.org/A221866 need some changes, should be: 1, 4, 57, 2473, 349712 (however, warning, 1x1 can't be played, because we need to place two 1) http://oeis.org/A224784 remain the same: 1, 4, 44, 2473, 297136 Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (8)
-
Allan Wechsler -
Christian Boyer -
Dan Asimov -
Eric Angelini -
Fred W. Helenius -
Hans Havermann -
Olivier Gerard -
Tom Rokicki