[math-fun] "Self-describing" integer tuples.
I am sure this has been discussed here before, so if somebody just has a pointer to any previous discussion, let me know. Suppose we have a finite sequence of non-negative integers. We can create a new sequence that is a "census" of the first. For example, the census of (1,4,1,4,2,1,3) is (3,1,1,2,1,3,2,4) -- that is, three 1's, one 2, one 3, and two 4's. I am interested in any tuple that is its own census. A simple example is (2,2); a more complicated example is (2,1,3,2,2,3,1,K). The K can be replaced with any number bigger than 3. I found another class of examples of the form (K,1,3,2,2,3,2,K,1,A,1,B...) where K,A,B... are distinct from each other and bigger than 3, and there are exactly K-1 terms in the A,B... set. Let's declare that the choice of the value of singletons doesn't matter. I have an intuition that the number of types of self-censusing sequences is quite small -- maybe only the three types I just mentioned? Or did I miss a few? I would not be surprised if I did, but I would be surprised if there were a _lot_ more types.
Allan,in the Index to the OEIS there is this entry self-describing numbers, sequences related to : [edit <https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit§ion=3> ]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101>, not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Oct 22, 2019 at 4:52 PM Allan Wechsler <acwacw@gmail.com> wrote:
I am sure this has been discussed here before, so if somebody just has a pointer to any previous discussion, let me know.
Suppose we have a finite sequence of non-negative integers. We can create a new sequence that is a "census" of the first. For example, the census of (1,4,1,4,2,1,3) is (3,1,1,2,1,3,2,4) -- that is, three 1's, one 2, one 3, and two 4's.
I am interested in any tuple that is its own census. A simple example is (2,2); a more complicated example is (2,1,3,2,2,3,1,K). The K can be replaced with any number bigger than 3.
I found another class of examples of the form (K,1,3,2,2,3,2,K,1,A,1,B...) where K,A,B... are distinct from each other and bigger than 3, and there are exactly K-1 terms in the A,B... set.
Let's declare that the choice of the value of singletons doesn't matter. I have an intuition that the number of types of self-censusing sequences is quite small -- maybe only the three types I just mentioned? Or did I miss a few? I would not be surprised if I did, but I would be surprised if there were a _lot_ more types. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dear Allan, Neil's link to the OEIS index leads to A047841 and A109776 which are close to your "census" map. I removed their tuples with zeroes, and with duplicates, sorted them by increasing digits, and checked (by a program) whether they are fixed points under the census map. If I got it right, then I think that the following ones marked with "*" are new: 2,1, 3,2, 2,3, 1,K (K>3) 2,2 3,1, 1,2, 3,3, 1,A (A>3) *3,1, 2,2, 3,3, 1,A, 1,B (A>3, B>A) *3,1, 3,3, 1,A, 1,B (A>3, B>A) 4,1, 3,2, 2,3, 2,4, 1,A, 1,B, 1,C (A>4, B>A, C>B *5,1, 3,2, 2,3, 1,4, 2,5, 1,A, 1,B, 1,C (A>5, B>A, C>B) 5,1, 3,2, 2,3, 2,5, 1,A, 1,B, 1,C, 1,D *6,1, 3,2, 2,3, 1,4, 1,5, 2,6, 1,7, 1,8, 1,9 I imagine that your definition is different from several existing OEIS sequences because it is base-independant and does not count zero. I can only speculate whether there are more "template" tuples. Regards - Georg Am 23.10.2019 um 18:45 schrieb Neil Sloane:
Allan,in the Index to the OEIS there is this entry
self-describing numbers, sequences related to : [edit <https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit§ion=3> ]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101>, not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil
I posted this query to math-fun and not to seqfan, because my intention was to collect a census of these tuples, not to suggest a new sequence. While there are sequences on OEIS that have a similar motivation, these sequences encrypt the basic concept by arbitrarily converting the tuples into base-10 integers. I think I already covered the concept "K1 32 23 2K 1A 1B 1C ...", which includes your asterisked 51 and 61 cases, unless I'm missing something. We can easily prove that "22" is the only one-pair example. There can be no two-pair examples. Such an example could only use two numbers, and the two possibilities "AA BB" and "AB BA" are easily seen to be impossible. I am pretty sure there can be no three-pair examples, though I don't have a fast proof. I am getting more and more convinced that we have now covered all the possible examples. Let's adopt the simplification of simply leaving all the singletons (the pairs of the form 1K) out of our presentations -- just assume the are off to the right. So the examples we have found have the following non-singleton parts: 22 21 32 23 31 33 K1 32 23 2K I am coming to the conclusion that this is all there is, and feel like a proof is "on the tip of my tongue." One lemma is: none of the numbers can exceed the number of pairs. On Thu, Oct 24, 2019 at 11:29 AM Georg Fischer <dr.georg.fischer@gmail.com> wrote:
Dear Allan,
Neil's link to the OEIS index leads to A047841 and A109776 which are close to your "census" map. I removed their tuples with zeroes, and with duplicates, sorted them by increasing digits, and checked (by a program) whether they are fixed points under the census map.
If I got it right, then I think that the following ones marked with "*" are new:
2,1, 3,2, 2,3, 1,K (K>3) 2,2 3,1, 1,2, 3,3, 1,A (A>3) *3,1, 2,2, 3,3, 1,A, 1,B (A>3, B>A) *3,1, 3,3, 1,A, 1,B (A>3, B>A) 4,1, 3,2, 2,3, 2,4, 1,A, 1,B, 1,C (A>4, B>A, C>B *5,1, 3,2, 2,3, 1,4, 2,5, 1,A, 1,B, 1,C (A>5, B>A, C>B) 5,1, 3,2, 2,3, 2,5, 1,A, 1,B, 1,C, 1,D *6,1, 3,2, 2,3, 1,4, 1,5, 2,6, 1,7, 1,8, 1,9
I imagine that your definition is different from several existing OEIS sequences because it is base-independant and does not count zero. I can only speculate whether there are more "template" tuples. Regards - Georg
Am 23.10.2019 um 18:45 schrieb Neil Sloane:
Allan,in the Index to the OEIS there is this entry
self-describing numbers, sequences related to : [edit < https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit&sec...
]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101 , not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil
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There is some relevant discussion in the "Variations" section of https://en.wikipedia.org/wiki/Look-and-say_sequence Andy On Thu, Oct 24, 2019 at 12:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I posted this query to math-fun and not to seqfan, because my intention was to collect a census of these tuples, not to suggest a new sequence. While there are sequences on OEIS that have a similar motivation, these sequences encrypt the basic concept by arbitrarily converting the tuples into base-10 integers.
I think I already covered the concept "K1 32 23 2K 1A 1B 1C ...", which includes your asterisked 51 and 61 cases, unless I'm missing something.
We can easily prove that "22" is the only one-pair example.
There can be no two-pair examples. Such an example could only use two numbers, and the two possibilities "AA BB" and "AB BA" are easily seen to be impossible.
I am pretty sure there can be no three-pair examples, though I don't have a fast proof.
I am getting more and more convinced that we have now covered all the possible examples. Let's adopt the simplification of simply leaving all the singletons (the pairs of the form 1K) out of our presentations -- just assume the are off to the right. So the examples we have found have the following non-singleton parts:
22 21 32 23 31 33 K1 32 23 2K
I am coming to the conclusion that this is all there is, and feel like a proof is "on the tip of my tongue." One lemma is: none of the numbers can exceed the number of pairs.
On Thu, Oct 24, 2019 at 11:29 AM Georg Fischer <dr.georg.fischer@gmail.com> wrote:
Dear Allan,
Neil's link to the OEIS index leads to A047841 and A109776 which are close to your "census" map. I removed their tuples with zeroes, and with duplicates, sorted them by increasing digits, and checked (by a program) whether they are fixed points under the census map.
If I got it right, then I think that the following ones marked with "*" are new:
2,1, 3,2, 2,3, 1,K (K>3) 2,2 3,1, 1,2, 3,3, 1,A (A>3) *3,1, 2,2, 3,3, 1,A, 1,B (A>3, B>A) *3,1, 3,3, 1,A, 1,B (A>3, B>A) 4,1, 3,2, 2,3, 2,4, 1,A, 1,B, 1,C (A>4, B>A, C>B *5,1, 3,2, 2,3, 1,4, 2,5, 1,A, 1,B, 1,C (A>5, B>A, C>B) 5,1, 3,2, 2,3, 2,5, 1,A, 1,B, 1,C, 1,D *6,1, 3,2, 2,3, 1,4, 1,5, 2,6, 1,7, 1,8, 1,9
I imagine that your definition is different from several existing OEIS sequences because it is base-independant and does not count zero. I can only speculate whether there are more "template" tuples. Regards - Georg
Am 23.10.2019 um 18:45 schrieb Neil Sloane:
Allan,in the Index to the OEIS there is this entry
self-describing numbers, sequences related to : [edit < https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit&sec...
]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101 , not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil
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-- Andy.Latto@pobox.com
Merci beaucoup ! à+ É. Catapulté de mon aPhone
Le 24 oct. 2019 à 19:29, Andy Latto <andy.latto@pobox.com> a écrit :
There is some relevant discussion in the "Variations" section of https://en.wikipedia.org/wiki/Look-and-say_sequence
Andy
On Thu, Oct 24, 2019 at 12:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I posted this query to math-fun and not to seqfan, because my intention was to collect a census of these tuples, not to suggest a new sequence. While there are sequences on OEIS that have a similar motivation, these sequences encrypt the basic concept by arbitrarily converting the tuples into base-10 integers.
I think I already covered the concept "K1 32 23 2K 1A 1B 1C ...", which includes your asterisked 51 and 61 cases, unless I'm missing something.
We can easily prove that "22" is the only one-pair example.
There can be no two-pair examples. Such an example could only use two numbers, and the two possibilities "AA BB" and "AB BA" are easily seen to be impossible.
I am pretty sure there can be no three-pair examples, though I don't have a fast proof.
I am getting more and more convinced that we have now covered all the possible examples. Let's adopt the simplification of simply leaving all the singletons (the pairs of the form 1K) out of our presentations -- just assume the are off to the right. So the examples we have found have the following non-singleton parts:
22 21 32 23 31 33 K1 32 23 2K
I am coming to the conclusion that this is all there is, and feel like a proof is "on the tip of my tongue." One lemma is: none of the numbers can exceed the number of pairs.
On Thu, Oct 24, 2019 at 11:29 AM Georg Fischer <dr.georg.fischer@gmail.com> wrote:
Dear Allan,
Neil's link to the OEIS index leads to A047841 and A109776 which are close to your "census" map. I removed their tuples with zeroes, and with duplicates, sorted them by increasing digits, and checked (by a program) whether they are fixed points under the census map.
If I got it right, then I think that the following ones marked with "*" are new:
2,1, 3,2, 2,3, 1,K (K>3) 2,2 3,1, 1,2, 3,3, 1,A (A>3) *3,1, 2,2, 3,3, 1,A, 1,B (A>3, B>A) *3,1, 3,3, 1,A, 1,B (A>3, B>A) 4,1, 3,2, 2,3, 2,4, 1,A, 1,B, 1,C (A>4, B>A, C>B *5,1, 3,2, 2,3, 1,4, 2,5, 1,A, 1,B, 1,C (A>5, B>A, C>B) 5,1, 3,2, 2,3, 2,5, 1,A, 1,B, 1,C, 1,D *6,1, 3,2, 2,3, 1,4, 1,5, 2,6, 1,7, 1,8, 1,9
I imagine that your definition is different from several existing OEIS sequences because it is base-independant and does not count zero. I can only speculate whether there are more "template" tuples. Regards - Georg
Am 23.10.2019 um 18:45 schrieb Neil Sloane: Allan,in the Index to the OEIS there is this entry
self-describing numbers, sequences related to : [edit < https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit&sec...
]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101 , not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil
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-- Andy.Latto@pobox.com
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Hello Math-Fun, Alice and Bob play the "Chunk & Sum" game. Which consist in A giving to B an integer and asking B to chunk it in smaller pieces that must be increasing from left to right. Example: 3223584222 is given by A to B. B chunks it like this: 3 22 358 4222. The "weight" of a partition is given by the sum of the pieces: 3 + 22 + 358 + 4222 = 4605 The aim of B is to produce the highest weight. The aim of A is to warn B, because his partition must obey a precise "minimal pieces" rule: "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 + 84 + 222 = 366". B agrees, his first partition forgot the "minimal pieces" way to chunk. After a while, Alice and Bob have an idea: to give to each other wooden blocks instead of integers -- blocks with numbers printed on them. A receives the blocks marked 1, 2, 3, 4, 5. B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The game consists now to put one's blocks in a certain order on the table such to produce a number to chunk – this number having the highest weight according to the two rules we have just seen: 1) pieces in increasing order from left to right 2) "minimal pieces". Alice forms the integer 43521 with her blocks and claims that among the 120 possible solutions, this is the best one. Indeed, the only way to properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you can reach. What is the integer Bob should form with his set of blocks in order to maximize the weight? ____________________ (warning: this is hard to compute by hand, my best result is over 100,000 -- took me the night) (note that the 1 and the 0 of the block marked 10 cannot be separated) This idea could produce a nice little sequence for the OEIS: "Highest possible weight that one can reach with blocks marked 1 to n, according to the Chunk & Sum game". I guess the sequence would start like this: 1, 21, 33, 325, 564,... (not in the OEIS, if the hereunder computations are ok) n = 1 --> integer produced is 1 with weight = 1 n = 2 --> integer produced is 21 with weight = 21 n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 etc. My solution for n = 10 is here, with Daniel Day-Lewis, on my personal blog: https://bit.ly/31KVedK) Best, É.
Please clarify the "minimal pieces" constraint? Everything seems to depend on that, and I cannot infer what it is from the examples given. On Sat, Oct 26, 2019, 6:16 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Alice and Bob play the "Chunk & Sum" game. Which consist in A giving to B an integer and asking B to chunk it in smaller pieces that must be increasing from left to right. Example: 3223584222 is given by A to B. B chunks it like this: 3 22 358 4222. The "weight" of a partition is given by the sum of the pieces: 3 + 22 + 358 + 4222 = 4605 The aim of B is to produce the highest weight. The aim of A is to warn B, because his partition must obey a precise "minimal pieces" rule: "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 + 84 + 222 = 366". B agrees, his first partition forgot the "minimal pieces" way to chunk.
After a while, Alice and Bob have an idea: to give to each other wooden blocks instead of integers -- blocks with numbers printed on them. A receives the blocks marked 1, 2, 3, 4, 5. B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The game consists now to put one's blocks in a certain order on the table such to produce a number to chunk – this number having the highest weight according to the two rules we have just seen: 1) pieces in increasing order from left to right 2) "minimal pieces". Alice forms the integer 43521 with her blocks and claims that among the 120 possible solutions, this is the best one. Indeed, the only way to properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you can reach. What is the integer Bob should form with his set of blocks in order to maximize the weight? ____________________ (warning: this is hard to compute by hand, my best result is over 100,000 -- took me the night) (note that the 1 and the 0 of the block marked 10 cannot be separated) This idea could produce a nice little sequence for the OEIS: "Highest possible weight that one can reach with blocks marked 1 to n, according to the Chunk & Sum game". I guess the sequence would start like this: 1, 21, 33, 325, 564,... (not in the OEIS, if the hereunder computations are ok) n = 1 --> integer produced is 1 with weight = 1 n = 2 --> integer produced is 21 with weight = 21 n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 etc. My solution for n = 10 is here, with Daniel Day-Lewis, on my personal blog: https://bit.ly/31KVedK) Best, É.
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You're right, Allan, I have problems in defining that rule -- which seems obvious when you work with it, but hard to explain (please someone, help!-) Here are a few examples, taken from my search of the "ten blocks" maximum weight: I started with 1,2,3,4,5,6,7,8,9,10 -- untouched, sum 55 then 10,9,8,7,6,5,4,3,2,1 -- blocks must be reorganized: try: 10, 98, 765, 4321 -- sum 5194 (I cannot reorganize in 109, 876, 5432, 1 as the "pieces" are not increasing from left to right -- BUT I cannot either reorganize in 10, 987654321 with sum 987654331 because Alice will tell me: "Too easy, pal! There is a way to break that in "minimal pieces!" And she would be right, of course with the solution above: 10, 98, 765, 4321 -- sum 5194. In short, Alice will always try to show me that the block ordering I've selected (among a huge lot of others -- factorial 10) can be "disassembled" to reduce the sum I was so proud of (the first constraint being still ok) Dunno if this helps... Best, É.
Le 26 octobre 2019 à 16:08, Allan Wechsler <acwacw@gmail.com> a écrit :
Please clarify the "minimal pieces" constraint? Everything seems to depend on that, and I cannot infer what it is from the examples given.
On Sat, Oct 26, 2019, 6:16 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Alice and Bob play the "Chunk & Sum" game. Which consist in A giving to B an integer and asking B to chunk it in smaller pieces that must be increasing from left to right. Example: 3223584222 is given by A to B. B chunks it like this: 3 22 358 4222. The "weight" of a partition is given by the sum of the pieces: 3 + 22 + 358 + 4222 = 4605 The aim of B is to produce the highest weight. The aim of A is to warn B, because his partition must obey a precise "minimal pieces" rule: "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 + 84 + 222 = 366". B agrees, his first partition forgot the "minimal pieces" way to chunk.
After a while, Alice and Bob have an idea: to give to each other wooden blocks instead of integers -- blocks with numbers printed on them. A receives the blocks marked 1, 2, 3, 4, 5. B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The game consists now to put one's blocks in a certain order on the table such to produce a number to chunk – this number having the highest weight according to the two rules we have just seen: 1) pieces in increasing order from left to right 2) "minimal pieces". Alice forms the integer 43521 with her blocks and claims that among the 120 possible solutions, this is the best one. Indeed, the only way to properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you can reach. What is the integer Bob should form with his set of blocks in order to maximize the weight? ____________________ (warning: this is hard to compute by hand, my best result is over 100,000 -- took me the night) (note that the 1 and the 0 of the block marked 10 cannot be separated) This idea could produce a nice little sequence for the OEIS: "Highest possible weight that one can reach with blocks marked 1 to n, according to the Chunk & Sum game". I guess the sequence would start like this: 1, 21, 33, 325, 564,... (not in the OEIS, if the hereunder computations are ok) n = 1 --> integer produced is 1 with weight = 1 n = 2 --> integer produced is 21 with weight = 21 n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 etc. My solution for n = 10 is here, with Daniel Day-Lewis, on my personal blog: https://bit.ly/31KVedK) Best, É.
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Another example, Allan: (same task -- "10 blocks") I was very proud of the integer 46385297110 that arises when you glue together the blocks 4.6.3.8.5.2.9.7.1.10 For me, this integer could only be chunked in 463+852+97110 sum 98029 Then came Alice (my feminine part) laughing: -- Ah, you've overlooked this "minimal pieces" arrangement: 4+6+38+52+97+110 sum (ridiculous) Well... only by swapping the first two blocks I got: 64385297110. And now, Alice, what do you say? Mmmmmh... 6+43+85+297+110 no (first rule) (let's start from the back with the "plus" signs): 6+4+38+52+97+110 no (first rule again) 6+438+529+7110 sum 8083... better but very low I had to search more and swap elsewhere: "Let's try this with me Alice please!" (if you can't beat them, join them): 65384297110 6+53+84+297110 = ... huge! But wait... 6+53+842+97110 is the "minimal pieces" solution, waow! sum = 98011 Etc. Best, É. (and A)
Le 26 octobre 2019 à 16:57, Éric Angelini <bk263401@skynet.be> a écrit :
You're right, Allan, I have problems in defining that rule -- which seems obvious when you work with it, but hard to explain (please someone, help!-)
Here are a few examples, taken from my search of the "ten blocks" maximum weight:
I started with 1,2,3,4,5,6,7,8,9,10 -- untouched, sum 55 then 10,9,8,7,6,5,4,3,2,1 -- blocks must be reorganized: try: 10, 98, 765, 4321 -- sum 5194 (I cannot reorganize in 109, 876, 5432, 1 as the "pieces" are not increasing from left to right -- BUT I cannot either reorganize in 10, 987654321 with sum 987654331 because Alice will tell me: "Too easy, pal! There is a way to break that in "minimal pieces!" And she would be right, of course with the solution above: 10, 98, 765, 4321 -- sum 5194. In short, Alice will always try to show me that the block ordering I've selected (among a huge lot of others -- factorial 10) can be "disassembled" to reduce the sum I was so proud of (the first constraint being still ok) Dunno if this helps... Best, É.
Le 26 octobre 2019 à 16:08, Allan Wechsler <acwacw@gmail.com> a écrit :
Please clarify the "minimal pieces" constraint? Everything seems to depend on that, and I cannot infer what it is from the examples given.
On Sat, Oct 26, 2019, 6:16 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Alice and Bob play the "Chunk & Sum" game. Which consist in A giving to B an integer and asking B to chunk it in smaller pieces that must be increasing from left to right. Example: 3223584222 is given by A to B. B chunks it like this: 3 22 358 4222. The "weight" of a partition is given by the sum of the pieces: 3 + 22 + 358 + 4222 = 4605 The aim of B is to produce the highest weight. The aim of A is to warn B, because his partition must obey a precise "minimal pieces" rule: "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 + 84 + 222 = 366". B agrees, his first partition forgot the "minimal pieces" way to chunk.
After a while, Alice and Bob have an idea: to give to each other wooden blocks instead of integers -- blocks with numbers printed on them. A receives the blocks marked 1, 2, 3, 4, 5. B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The game consists now to put one's blocks in a certain order on the table such to produce a number to chunk – this number having the highest weight according to the two rules we have just seen: 1) pieces in increasing order from left to right 2) "minimal pieces". Alice forms the integer 43521 with her blocks and claims that among the 120 possible solutions, this is the best one. Indeed, the only way to properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you can reach. What is the integer Bob should form with his set of blocks in order to maximize the weight? ____________________ (warning: this is hard to compute by hand, my best result is over 100,000 -- took me the night) (note that the 1 and the 0 of the block marked 10 cannot be separated) This idea could produce a nice little sequence for the OEIS: "Highest possible weight that one can reach with blocks marked 1 to n, according to the Chunk & Sum game". I guess the sequence would start like this: 1, 21, 33, 325, 564,... (not in the OEIS, if the hereunder computations are ok) n = 1 --> integer produced is 1 with weight = 1 n = 2 --> integer produced is 21 with weight = 21 n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 etc. My solution for n = 10 is here, with Daniel Day-Lewis, on my personal blog: https://bit.ly/31KVedK) Best, É.
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I'm afraid I am still baffled. You still never define the "minimal pieces" constraint. Bob is trying to maximize the sum, and produces a chunking with a great big sum. Alice says, "No, no!" and counters with a different chunking with a smaller sum. It is obvious why Bob prefers his choice: his sum is bigger. It is NOT obvious why Alice prefers hers. At this point, the "minimal pieces" constraint is always mentioned to explain Alice's demurral, but even with a half dozen examples in front of me, I cannot guess what that constraint is.I admit this may be because I am not intelligent enough. But -- they say you don't know how to explain something until you can explain it to the unintelligent. On Sat, Oct 26, 2019 at 11:22 AM Éric Angelini <bk263401@skynet.be> wrote: > Another example, Allan: > (same task -- "10 blocks") > > I was very proud of the integer 46385297110 > that arises when you glue together the blocks > 4.6.3.8.5.2.9.7.1.10 > For me, this integer could only be chunked in > 463+852+97110 sum 98029 > Then came Alice (my feminine part) laughing: > -- Ah, you've overlooked this "minimal pieces" > arrangement: > 4+6+38+52+97+110 sum (ridiculous) > Well... only by swapping the first two blocks I got: > 64385297110. > And now, Alice, what do you say? > Mmmmmh... > 6+43+85+297+110 no (first rule) > (let's start from the back with the "plus" signs): > 6+4+38+52+97+110 no (first rule again) > 6+438+529+7110 sum 8083... better but very low > I had to search more and swap elsewhere: > "Let's try this with me Alice please!" (if you can't > beat them, join them): > 65384297110 > 6+53+84+297110 = ... huge! But wait... > 6+53+842+97110 is the "minimal pieces" solution, waow! > sum = 98011 > Etc. > Best, > É. > (and A) > > > > > > > Le 26 octobre 2019 à 16:57, Éric Angelini <bk263401@skynet.be> a écrit : > > > > > > You're right, Allan, I have problems in defining that rule -- > > which seems obvious when you work with it, but hard to explain > > (please someone, help!-) > > > > Here are a few examples, taken from my search of the "ten blocks" > > maximum weight: > > > > I started with 1,2,3,4,5,6,7,8,9,10 -- untouched, sum 55 > > then 10,9,8,7,6,5,4,3,2,1 -- blocks must be reorganized: > > try: 10, 98, 765, 4321 -- sum 5194 > > (I cannot reorganize in 109, 876, 5432, 1 as the "pieces" are > > not increasing from left to right -- BUT I cannot either > > reorganize in 10, 987654321 with sum 987654331 because > > Alice will tell me: "Too easy, pal! There is a way to break > > that in "minimal pieces!" And she would be right, of course > > with the solution above: 10, 98, 765, 4321 -- sum 5194. > > In short, Alice will always try to show me that the block > > ordering I've selected (among a huge lot of others -- > > factorial 10) can be "disassembled" to reduce the sum I was > > so proud of (the first constraint being still ok) > > Dunno if this helps... > > Best, > > É. > > > > > > > > > > > > > > > Le 26 octobre 2019 à 16:08, Allan Wechsler <acwacw@gmail.com> a > écrit : > > > > > > > > > Please clarify the "minimal pieces" constraint? Everything seems to > depend > > > on that, and I cannot infer what it is from the examples given. > > > > > > On Sat, Oct 26, 2019, 6:16 AM Éric Angelini <bk263401@skynet.be> > wrote: > > > > > > > Hello Math-Fun, > > > > Alice and Bob play the "Chunk & Sum" game. > > > > Which consist in A giving to B an integer and asking B to chunk it in > > > > smaller pieces that must be increasing from left to right. > > > > Example: > > > > 3223584222 is given by A to B. > > > > B chunks it like this: 3 22 358 4222. > > > > The "weight" of a partition is given by the sum of the pieces: 3 + > 22 + > > > > 358 + 4222 = 4605 > > > > The aim of B is to produce the highest weight. > > > > The aim of A is to warn B, because his partition must obey a precise > > > > "minimal pieces" rule: > > > > "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 > + 84 > > > > + 222 = 366". > > > > B agrees, his first partition forgot the "minimal pieces" way to > chunk. > > > > > > > > After a while, Alice and Bob have an idea: to give to each other > wooden > > > > blocks instead of integers -- blocks with numbers printed on them. > > > > A receives the blocks marked 1, 2, 3, 4, 5. > > > > B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. > > > > The game consists now to put one's blocks in a certain order on the > table > > > > such to produce a number to chunk – this number having the highest > weight > > > > according to the two rules we have just seen: > > > > 1) pieces in increasing order from left to right > > > > 2) "minimal pieces". > > > > Alice forms the integer 43521 with her blocks and claims that among > the > > > > 120 possible solutions, this is the best one. Indeed, the only way to > > > > properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you > can reach. > > > > What is the integer Bob should form with his set of blocks in order > to > > > > maximize the weight? > > > > ____________________ > > > > (warning: this is hard to compute by hand, my best result is over > 100,000 > > > > -- took me the night) > > > > (note that the 1 and the 0 of the block marked 10 cannot be > separated) > > > > This idea could produce a nice little sequence for the OEIS: "Highest > > > > possible weight that one can reach with blocks marked 1 to n, > according to > > > > the Chunk & Sum game". I guess the sequence would start like this: > 1, 21, > > > > 33, 325, 564,... (not in the OEIS, if the hereunder computations are > ok) > > > > n = 1 --> integer produced is 1 with weight = 1 > > > > n = 2 --> integer produced is 21 with weight = 21 > > > > n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 > > > > n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 > > > > n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 > > > > etc. > > > > My solution for n = 10 is here, with Daniel Day-Lewis, on my > personal blog: > > > > https://bit.ly/31KVedK) > > > > Best, > > > > É. > > > > > > > > _______________________________________________ > > > > math-fun mailing list > > > > math-fun@mailman.xmission.com > > > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > > _______________________________________________ > > > math-fun mailing list > > > math-fun@mailman.xmission.com > > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
Hello Math-Fun, I read here https://bit.ly/2NmIOnh that 31,415,926,535,897 base-10 digits of pi are now known. Does it mean that if we have a list of, say, 10,000 digits of pi that starts with the 40,000,000,000,000th digit of pi, we won't recognize it as coming from pi (secretely computed so far by someone)? I guess we won't -- as pi digits look random. What about a 10,000 digits list of zeros and ones starting with the 40,000,000,000,000th digit of the Kolakoski seq? (again, let's imagine someone has secretely computed that). Is there a test that could give us a hint (the difference with pi being that Kolakoski is a selfdescribing sequence)? Thanks, É.
Hello, the answer is no. First the Kolakoski sequence (A000002) has a specific binary pattern, see for yourself here : http://plouffe.fr/100%20millions%20de%20valeurs%20Kolakoski.png there are 100 million values of A000002 encoded in a color image (for visibility), as you can see if we compare to an image of let's say 100 million digits (in binary) of Pi : there is no doubt about it : Pi is random, Kolakoski is not, it is bizarre, strange and such, but surely not the same as a pure random sequence. Some people I know said me once that this sequence will turn you crazy (la suite qui rend fou). Best regards, bonne soirée. Simon Plouffe Le 2019-10-24 à 18:48, Éric Angelini a écrit :
Hello Math-Fun, I read here https://bit.ly/2NmIOnh that 31,415,926,535,897 base-10 digits of pi are now known.
Does it mean that if we have a list of, say, 10,000 digits of pi that starts with the 40,000,000,000,000th digit of pi, we won't recognize it as coming from pi (secretely computed so far by someone)? I guess we won't -- as pi digits look random.
What about a 10,000 digits list of zeros and ones starting with the 40,000,000,000,000th digit of the Kolakoski seq? (again, let's imagine someone has secretely computed that). Is there a test that could give us a hint (the difference with pi being that Kolakoski is a selfdescribing sequence)?
Thanks, É.
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participants (7)
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Allan Wechsler -
Andy Latto -
Georg Fischer -
Neil Sloane -
Simon Plouffe -
Éric Angelini -
Éric Angelini