Enigma 531: Petits fours
23 December 2019
Posted by on
From New Scientist #1683, 23rd September 1989 [link]
“Four-armed is four-warmed,” declared Professor Törqui 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 10¹, poor thing) are expressible as sums of strings of 4s.”
“The most economical way of expressing 10² as a sum of strings of 4s (that is, the one using fewest strings and hence fewest 4s) uses seven 4s:”
10² = 44 + 44 + 4 + 4 + 4.
“The most economical means of expressing 10³ as a sum of strings of 4s requires sixteen 4s:”
10³ = 444 + 444 + 44 + 44 + 4 + 4 + 4 + 4 + 4 + 4.
“Now, it’s four o’clock, and just time for this puzzle: Give me somewhere to put my cakestand 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 Törqui’s boast would commit him to baking? (Express your answer as a power of 10.)
[*] £44.44 from Whatsit Forum.