Spoiler for New Scientist Enigma 1701:
The display on my calculator shows 9876543210. As usual, up to seven illuminated strips are used to display each digit – the 8 using all seven, for example. There is just one special 10-figure number with the property that it is a perfect power of the total number of illuminated strips that it uses.
With a little calculator effort it is possible to answer the following: How many illuminated strips does this special 10-figure number use?
I don’t see any particularly clever way to do this. Well, you can narrow the solution space some. The digits from 0 to 9 have 6, 2, 5, 5, 4, 5, 6, 3, 7, 5 illuminated segments, respectively. The number of illuminated segments in total must be at least 20 (1111111111) and at most 70 (8888888888). Of the numbers in that range, 20 through 26 have 10-digit 7th powers, 32 through 46 have 10-digit 6th powers, and 64 through 70 have 10-digit 5th powers. But numbers using fewer than 40 segments — fewer than 4, on average, per digit — would have to have lots of 1s and/or 7s, but none of the 10-digit powers of 20–26 and 32–39 does. Similarly, numbers using more than 64 segments — more than 6, on average, per digit — would have to use a lot of 8s, but none of the 10-digit powers of 64–70 does. And 40^6 = 4,096,000,000, which uses 42 segments in its zeroes alone. So the solution presumably must be in the range 41 through 46.
From there it’s just checking every possibility until you find the solution. I started at 46 and worked down, which was the wrong direction. The 10-digit number is 4,750,104,241, using 4+3+5+6+2+6+4+5+4+2=41 segments, and 41^6 = 4,750,104,241.
I don’t see what the first sentence of the problem statement has to do with anything, unless it’s to tell you commas are not displayed.