The Universe of Discourse


Tue, 13 Nov 2018

A puzzle about representing numbers as a sum of 3-smooth numbers

I think this would be fun for a suitably-minded bright kid of maybe 12–15 years old.

Consider the following table of numbers of the form !!2^i3^j!!:

1 3 9 27 81 243
2 6 18 54 162
4 12 36 108
8 24 72 216
16 48 144
32 96
64 192
128

Given a number !!n!!, is is possible to represent !!n!! as a sum of entries from the table, with the following constraints:

  • No more than one entry from any column
  • An entry may only be used if it is in a strictly higher row than any entry to its left.

For example, one may not represent !!23 = 2 + 12 + 9!!, because the !!12!! is in a lower row than the !!2!! to its left.

1 3 9 27
2 6 18 54
4 12 36 108

But !!23 = 8 + 6 + 9!! is acceptable, because 6 is higher than 8, and 9 is higher than 6.

1 3 9 27
2 6 18 54
4 12 36 108
8 24 72 216

Or, put another way: can we represent any number !!n!! in the form $$n = \sum_i 2^{a_i}3^{b_i}$$ where the !!a_i!! are strictly decreasing and the !!b_i!! are strictly increasing?

Spoiler:

maxpow3 1 = 1 maxpow3 2 = 1 maxpow3 n = 3 * maxpow3 (n `div` 3) rep :: Integer -> [Integer] rep 0 = [] rep n = if even n then map (* 2) (rep (n `div` 2)) else (rep (n - mp3)) ++ [mp3] where mp3 = maxpow3 n

Sadly, the representation is not unique. For example, !!8+3 = 2+9!!, and !!32+24+9 = 32+6+27 = 8+12=18+27!!.


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