There's a cute puzzle at https://www.foxtrot.com/2019/01/13/cell-division/ I get a unique answer. Perhaps the smallest such puzzle? Rich
Agreed; but smallest in what metric? WFL On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
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I dunno -- fewest symbols? smallest dividend? --Rich Quoting Fred Lunnon <fred.lunnon@gmail.com>:
Agreed; but smallest in what metric? WFL
On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Oops, I misread Rich’s question as perhaps smallest *solution* to this specific puzzle, not smallest *puzzle*. Smallest *puzzle* is a more interesting question, but here’s what I found anyhow. Maybe smallest radix? I find a parametric solution, good for n=1, 2, 3, … With the variables named as diagrammed below, and assuming they must be distinct digits, A = 2n B = n(4n+1) C = n(4n+2) D = 0 E = n radix = 2n(4n+1) = 10, 36, 78, 136, 210, 300, ... Per program search, this is the only solution through radix=320, but I have no proof there aren’t others in higher radix. A C ------- A B ) C B D B D --- E B D E B D ----- D — Mike
On Jan 14, 2019, at 12:41 AM, rcs@xmission.com wrote:
I dunno -- fewest symbols? smallest dividend? --Rich
Quoting Fred Lunnon <fred.lunnon@gmail.com>:
Agreed; but smallest in what metric? WFL
On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
OK, now addressing Rich’s actual question. There are 601 puzzles (division problems) that have the “correct form”. Each has 5 to 10 symbols; so 5 is the fewest symbols (as in Foxtrot), and 40 of the 601 have 5 symbols. Among the 40 puzzles, there are 31 patterns of where those 5 symbols appear; 22 patterns have a unique solution, and 9 have two solutions. Among the 40, the divisor ranges from 228 to 960. The smallest divisor is 228, and it occurs with only one pattern, so its solution is unique. That puzzle is: A C --------------- A B ) B B D A B ----- A Z D A Z D ----- Z (“Correct form” = radix ten, 2-digit divisor, 2-digit quotient, 3-digit dividend, 2-digit first product, 3-digit second product, no leading zeros, remainder zero.) — Mike
On Jan 14, 2019, at 12:41 AM, rcs@xmission.com wrote:
I dunno -- fewest symbols? smallest dividend? --Rich
Quoting Fred Lunnon <fred.lunnon@gmail.com>:
Agreed; but smallest in what metric? WFL
On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
"The smallest dividend is 228" perhaps? WFL On 1/15/19, Mike Beeler <mikebeeler2@gmail.com> wrote:
OK, now addressing Rich’s actual question. There are 601 puzzles (division problems) that have the “correct form”. Each has 5 to 10 symbols; so 5 is the fewest symbols (as in Foxtrot), and 40 of the 601 have 5 symbols. Among the 40 puzzles, there are 31 patterns of where those 5 symbols appear; 22 patterns have a unique solution, and 9 have two solutions. Among the 40, the divisor ranges from 228 to 960. The smallest divisor is 228, and it occurs with only one pattern, so its solution is unique. That puzzle is:
A C --------------- A B ) B B D A B ----- A Z D A Z D ----- Z
(“Correct form” = radix ten, 2-digit divisor, 2-digit quotient, 3-digit dividend, 2-digit first product, 3-digit second product, no leading zeros, remainder zero.)
— Mike
On Jan 14, 2019, at 12:41 AM, rcs@xmission.com wrote:
I dunno -- fewest symbols? smallest dividend? --Rich
Quoting Fred Lunnon <fred.lunnon@gmail.com>:
Agreed; but smallest in what metric? WFL
On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Oops, yes, the smallest dividend is 228. And that’s the smallest not just among the 40 that have 5 different symbols, but smallest among all 601 puzzles of the Foxtrot form.
On Jan 14, 2019, at 11:53 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
"The smallest dividend is 228" perhaps? WFL
On 1/15/19, Mike Beeler <mikebeeler2@gmail.com> wrote:
OK, now addressing Rich’s actual question. There are 601 puzzles (division problems) that have the “correct form”. Each has 5 to 10 symbols; so 5 is the fewest symbols (as in Foxtrot), and 40 of the 601 have 5 symbols. Among the 40 puzzles, there are 31 patterns of where those 5 symbols appear; 22 patterns have a unique solution, and 9 have two solutions. Among the 40, the divisor ranges from 228 to 960. The smallest divisor is 228, and it occurs with only one pattern, so its solution is unique. That puzzle is:
A C --------------- A B ) B B D A B ----- A Z D A Z D ----- Z
(“Correct form” = radix ten, 2-digit divisor, 2-digit quotient, 3-digit dividend, 2-digit first product, 3-digit second product, no leading zeros, remainder zero.)
— Mike
On Jan 14, 2019, at 12:41 AM, rcs@xmission.com wrote:
I dunno -- fewest symbols? smallest dividend? --Rich
Quoting Fred Lunnon <fred.lunnon@gmail.com>:
Agreed; but smallest in what metric? WFL
On 1/13/19, rcs@xmission.com <rcs@xmission.com> wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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One last twist on the Foxtrot long division alphametric puzzle. The original has 16 digit positions. I think it can be reduced to 14, and still have a non-trivial puzzle that illustrates the operations in long division. I suggest this "mini-Foxtrot" puzzle: (fixed width font needed) C D 2 4 ---------- ---------- A B ) B A C 1 3 ) 3 1 2 C E 2 6 ---- ---- F C 5 2 F C 5 2 ---- ---- Z 0 In looking for puzzles with few digit positions, a lot of simple cases arise, such as "121 / 11 = 11", "144 / 12 = 12", etc. That made me wonder what might be the *hardest* puzzles. I suspect those that use all ten digits are hardest. Below are the 4 of original Foxtrot structure, and the 2 of mini-Foxtrot. Each has a unique solution. C D ---------- A B ) E F G H I ---- A A G A A G ------- Z C D ---------- A B ) E Z F G H ---- A I F A I F ------- Z C D ---------- A B ) E F C G H ---- F I C F I C ------- Z C D ---------- A B ) B A E F G ---- H I E H I E ------- Z C D ---------- A B ) E F G E H ---- I G I G ---- Z C D ---------- A B ) D E F G H ---- I F I F ---- Z SPOILER ALERT 592 / 16 = 37 702 / 18 = 39 912 / 38 = 24 936 / 39 = 24 598 / 13 = 46 456 / 19 = 24 -- Mike
On Jan 13, 2019, at 5:39 PM, rcs@xmission.com wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
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Mike: Perhaps this is already known, but what about long division puzzles which have unique solutions *for each radix*, but can be (uniquely) solved for multiple radices (radishes?) ? BTW, one of the early claims for the Prolog programming language was its ability to "trivially" solve digit arithmetic puzzles; Prolog automatically performed "hypothesize & test" for you, with builtin backtracking when an hypothesis failed. At 10:41 AM 1/18/2019, Mike Beeler wrote:
One last twist on the Foxtrot long division alphametric puzzle. The original has 16 digit positions. I think it can be reduced to 14, and still have a non-trivial puzzle that illustrates the operations in long division. I suggest this "mini-Foxtrot" puzzle: (fixed width font needed)
C D 2 4 ---------- ---------- A B ) B A C 1 3 ) 3 1 2 C E 2 6 ---- ---- F C 5 2 F C 5 2 ---- ---- Z 0
In looking for puzzles with few digit positions, a lot of simple cases arise, such as "121 / 11 = 11", "144 / 12 = 12", etc. That made me wonder what might be the *hardest* puzzles. I suspect those that use all ten digits are hardest. Below are the 4 of original Foxtrot structure, and the 2 of mini-Foxtrot. Each has a unique solution.
C D ---------- A B ) E F G H I ---- A A G A A G ------- Z
C D ---------- A B ) E Z F G H ---- A I F A I F ------- Z
C D ---------- A B ) E F C G H ---- F I C F I C ------- Z
C D ---------- A B ) B A E F G ---- H I E H I E ------- Z
C D ---------- A B ) E F G E H ---- I G I G ---- Z
C D ---------- A B ) D E F G H ---- I F I F ---- Z
SPOILER ALERT 592 / 16 = 37 702 / 18 = 39 912 / 38 = 24 936 / 39 = 24 598 / 13 = 46 456 / 19 = 24
-- Mike
On Jan 13, 2019, at 5:39 PM, rcs@xmission.com wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
For definiteness, I take Henry's question to be, "Is there a puzzle with the original Foxtrot structure, that has a unique solution in every radix starting with some threshold?" I found these five: # dif syms symbol pattern divisor quotient radix 5 ABACBBDABAZ 1 2 1 r-1 5..100 6 AABCDBEAABC 2 2 1 r-1 7..100 6 ABCDBCEABCD 2 4 1 r-3 17..100 (and 15) 7 ABCDBEFABAC 2 4 1 r-1 10..100 (and 7) 7 ABCDEEFABAC 3 5 1 r-1 12..100 (and 9) Reminder: the symbol pattern is the symbols for digit positions p1 through p11, in order. Symbol Z=0 is implicitly present, even if it does not occur in p1 through p11. The original Foxtrot structure is: p3 p4 ------------ p1 p2 ) p5 p6 p7 p8 p9 ------- p10 p11 p7 p10 p11 p7 ---------- Z The first symbol pattern above, ABACBBDABAZ, makes: A C ---------- A B ) B B D A B ---- A Z D A Z D ------- Z And that has the unique solution, for radix 5 through 100: 1 r-1 ------------ 1 2 ) 2 2 r-2 1 2 ---- 1 0 r-2 1 0 r-2 --------- 0 A short form to speccify this puzzle and solution is: (2 2 r-2) / (1 2) = (1 r-1) (radix r = 5 to at least 100) Existence of a unique solution for each symbol pattern was checked through radix=100. For the first two and last two, I think I proved a unique solution for all r; but a proof eludes me for the third one. That worries me, because another pattern, ABACDEFEDGC, has a unique solution (divisor = 2 3, quotient = 2 r-2) for radix 13..52; but for radix 53 it has a second solution (divisor = 6 34, quotient = 6 47)! — Mike
On Jan 18, 2019, at 2:55 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Mike:
Perhaps this is already known, but what about long division puzzles which have unique solutions *for each radix*, but can be (uniquely) solved for multiple radices (radishes?) ?
BTW, one of the early claims for the Prolog programming language was its ability to "trivially" solve digit arithmetic puzzles; Prolog automatically performed "hypothesize & test" for you, with builtin backtracking when an hypothesis failed.
On Jan 13, 2019, at 5:39 PM, rcs@xmission.com wrote:
There's a cute puzzle at
https://www.foxtrot.com/2019/01/13/cell-division/
I get a unique answer. Perhaps the smallest such puzzle?
Rich
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