Comments on: Euler Problem 112 http://www.dailydoseofexcel.com/archives/2009/07/11/euler-problem-112/ Daily posts of Excel tips…and other stuff Wed, 08 Feb 2012 23:58:05 +0000 hourly 1 http://wordpress.org/?v=3.2.1 By: fzz http://www.dailydoseofexcel.com/archives/2009/07/11/euler-problem-112/#comment-40225 fzz Sun, 12 Jul 2009 10:45:00 +0000 http://www.dailydoseofexcel.com/?p=2707#comment-40225 <p>dermotb is on the right track. Bounciness is recursive. Let letters be decimal digits, then if cba is bouncy, so is dcba, edcba, fedcba, etc as well as cbaz, cbazy, cbazyx etc. And so are abc, abcd, abcde, abcdef, zabc, yzabc, xyzabc, etc.</p> <p>Actually, given abc distinct digits with a less than b less than c, the permutations acb, bac, bca and cab are bouncy, so 2/3 of 3-digit numbers with 3 distinct digits are bouncy. With 2 distinct numbers aab, the permutation aba is bouncy while aab and baa aren't, so 1/3 of 3-digit numbers with 2 distinct digits are bouncy. No 3-digit numbers with all digits the same are bouncy.</p> <p>Generalize to permutations of more numbers.</p> dermotb is on the right track. Bounciness is recursive. Let letters be decimal digits, then if cba is bouncy, so is dcba, edcba, fedcba, etc as well as cbaz, cbazy, cbazyx etc. And so are abc, abcd, abcde, abcdef, zabc, yzabc, xyzabc, etc.

Actually, given abc distinct digits with a less than b less than c, the permutations acb, bac, bca and cab are bouncy, so 2/3 of 3-digit numbers with 3 distinct digits are bouncy. With 2 distinct numbers aab, the permutation aba is bouncy while aab and baa aren’t, so 1/3 of 3-digit numbers with 2 distinct digits are bouncy. No 3-digit numbers with all digits the same are bouncy.

Generalize to permutations of more numbers.

]]> By: dermotb http://www.dailydoseofexcel.com/archives/2009/07/11/euler-problem-112/#comment-40221 dermotb Sun, 12 Jul 2009 01:20:00 +0000 http://www.dailydoseofexcel.com/?p=2707#comment-40221 <p>Michael, one answer may be to skip big chunks of numbers. For example, if you are looking at the number 24510000, then clearly the next 45554 numbers are going to be bouncy, until you get to 24155555. It may take some twisted logic to achieve, but would save heaps of time.</p> Michael, one answer may be to skip big chunks of numbers. For example, if you are looking at the number 24510000, then clearly the next 45554 numbers are going to be bouncy, until you get to 24155555. It may take some twisted logic to achieve, but would save heaps of time.

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