New Scientist Enigma 531 – Petits Fours

by Chris Maslanka

From Issue #1683, 23rd September 1989

“Four-armed is four-warmed,” declared Professor Torqui as he placed the petits fours in the oven in his lab at the Department of Immaterial Science and Unclear Physics.

“There are 4444 of them: a string of 4s. By which I mean, naturally enough, a number in base 10 all of whose digits are 4. Do you like my plus fours? (*) Speaking of 10s and plus fours, you can hardly be unaware of the fact that all positive integral powers of 10 (except 10^1, poor thing) are expressible as sums of strings of 4s.

The most economical way of expressing 10^2 as a sum of strings of 4s (that is, the one using fewest strings and hence fewest 4s) uses seven 4s:

10^2 = 44 + 44 + 4 + 4 + 4

The most economical means of expressing 10^3 as a sum of strings of 4s requires sixteen 4s:

10^3 = 444 + 444 + 44 + 44 + 4 + 4 + 4 + 4 + 4 + 4

“Now, it’s four o’clock, and just time for this puzzle: Give me some-where to put my cake stand and I will make a number of petits fours which is an integral positive power of 10 such that the number of 4s required to write it as a sum of strings of 4s in the most economical way is itself a string of 4s.”

What is the smallest number of petits fours Torqui’s boast would commit him to baking? (Express your answer as a power of 10).

(*) £44-44 from Whatsit Forum.