[math-fun] A generalization of an old Conway Problem
John Conway once posed the following problem: The Ten Divisibilities I have a ten digit number, abcdefghij. Each of the digits is different, and a is divisible by 1 ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible by 9 abcdefghij is divisible by 10 Find the number. One can do this computationally in a number of ways, such as exhausting over all permutations of [1,10], or branch and bound. I decided to do this via integer programming to investigate the generalization to other bases than 10. Interestingly enough, I found that there were only solutions for bases 2,4,6,8,10,14 (tested through base 27). I don't know if they've petered out completely. Any ideas? Victor
Hans, Thanks. On Mon, May 25, 2020 at 17:12 Hans Havermann <gladhobo@bell.net> wrote:
http://oeis.org/A111456 http://oeis.org/A256112
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a is divisible by 1 ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible by 9 abcdefghij is divisible by 10 http://oeis.org/A111456 http://oeis.org/A256112 I was initially confused about the difference between these two sequences. The second one relaxes the divisibility criteria slightly by not insisting that the entire number be divisible by the base. If it is divisible by the base, that base is even and the number (in BaseForm) ends in 0. Otherwise, it appears that for even bases the final digit is base-1 and for odd bases, (base-1)/2. See: http://chesswanks.com/num/a256112.pdf For bases > 10, the convention a=10, b=11, c=12, .. is used. The ten A111456 terms are highlighted.
Base pandigital abcd... ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible by 9 abcdefghij is divisible by 10 abcdefghijk is divisible by 11 ... up to "... is divisible by base-1" I've just added base-37 to my A256112 < http://chesswanks.com/num/a256112.pdf > solutions list. I needed to borrow a letter from the Greek alphabet! Can someone explain why prime bases have so many more solutions than composite ones?
Upper case Roman (& Greek)? Russian? WFL On 6/4/20, Hans Havermann <gladhobo@bell.net> wrote:
Base pandigital abcd... ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible by 9 abcdefghij is divisible by 10 abcdefghijk is divisible by 11 ... up to "... is divisible by base-1"
I've just added base-37 to my A256112 < http://chesswanks.com/num/a256112.pdf > solutions list. I needed to borrow a letter from the Greek alphabet! Can someone explain why prime bases have so many more solutions than composite ones? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Il 04/06/2020 02:06, Hans Havermann ha scritto:
I've just added base-37 to my A256112 < http://chesswanks.com/num/a256112.pdf > solutions list. I needed to borrow a letter from the Greek alphabet! Can someone explain why prime bases have so many more solutions than composite ones? When the base B is composite and you are selecting the k-th digit, where k is a divisor of B, you have less choices, because that digit must be a divisor of B as well.
So, for example, if B is even, you can only use even digits in even positions, reducing the number of possibilities. (The last digit does not follow this rule because the whole number does not need to be a multiple of B). If B is prime you have fewer constraints. Giovanni
http://chesswanks.com/num/a256112.pdf Giovanni Resta has extended the sequence allowing me to update the pdf.
participants (4)
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Fred Lunnon -
Giovanni Resta -
Hans Havermann -
Victor Miller