- #1
SSequence
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Here is a a description of the problem:
http://mathcentral.uregina.ca/QQ/database/QQ.09.99/jackson2.html
http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/shanna1.html
The question is also in one of Martin Gardner's puzzle books. I was thinking about the solution of the problem for a general input (any non-negative integer).
It is reasonably clear that for large enough values, eventually there are going to be input values where a switch to a higher base would be made (in the solution --- using minimum number of pieces).
Take the example of 999:
100 length --- 9 pieces
10 length --- 9 pieces
1 length --- 9 pieces
This division is likely not the solution for 999 but as we increase the input (say 9999) this probably becomes a better approximation to the solution. And eventually higher and higher bases (and possibly some combinations of higher and lower bases too?) would be used in solutions.
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But there is also the question of an elegant (relatively easy to describe) and reasonably efficient method for values that don't admit a direct solution.
For example, input values like 31,63, etc. have direct solutions (it seems "apparently" that eventually all base values will be used in at least one direct solution).
There is also the question of input values for which there are more than one solutions (that is, two different divisions that use the same number of pieces). Can one find an example of their existence ... or disprove their existence mathematically.
http://mathcentral.uregina.ca/QQ/database/QQ.09.99/jackson2.html
http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/shanna1.html
The question is also in one of Martin Gardner's puzzle books. I was thinking about the solution of the problem for a general input (any non-negative integer).
It is reasonably clear that for large enough values, eventually there are going to be input values where a switch to a higher base would be made (in the solution --- using minimum number of pieces).
Take the example of 999:
100 length --- 9 pieces
10 length --- 9 pieces
1 length --- 9 pieces
This division is likely not the solution for 999 but as we increase the input (say 9999) this probably becomes a better approximation to the solution. And eventually higher and higher bases (and possibly some combinations of higher and lower bases too?) would be used in solutions.
=====
But there is also the question of an elegant (relatively easy to describe) and reasonably efficient method for values that don't admit a direct solution.
For example, input values like 31,63, etc. have direct solutions (it seems "apparently" that eventually all base values will be used in at least one direct solution).
There is also the question of input values for which there are more than one solutions (that is, two different divisions that use the same number of pieces). Can one find an example of their existence ... or disprove their existence mathematically.