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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!!:
Given a number !!n!!, is is possible to represent !!n!! as a sum of entries from the table, with the following constraints:
For example, one may not represent !!23 = 2 + 12 + 9!!, because the !!12!! is in a lower row than the !!2!! to its left.
But !!23 = 8 + 6 + 9!! is acceptable, because 6 is higher than 8, and 9 is higher than 6.
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:
Sadly, the representation is not unique. For example, !!8+3 = 2+9!!, and !!32+24+9 = 32+6+27 = 8+12=18+27!!. [Other articles in category /math] permanent link |