Enigma Number 1628

Posted on by Chris Warburton

Was reading through New Scientist and thought I’d have a go at their “Enigma” puzzle, which I’ve always glossed over since they’re pretty difficult maths. However, since I’ve got a computer I thought I’d see what the answer is anyway.

The question is this:

Given this, what is the 2 digit number x such that:

Translating this into Python is quite simple. We start with all of the possible answers, ie. the numbers 00-99, which we get via range(100).

Next we can split them into a pair of the “tens and units” as they used to say in primary school. We do this with [(a/10,a%10) for a in range(100)].

Next we can put these back into a list of numbers by undoing the separation of the digits, like this [x*10+y for x,y in [(a/10,a%10) for a in range(100)]].

Next we want to put our filters on this. Namely, the 2 digits x and y cannot be equal, so x!=y; the number raised to itself, (x*10+y)**(x*10+y), must ends with itself, ie. ((x*10+y)**(x*10+y))%100==x*10+y; the same must be true for the reverse, where we just swap x and y; and the reverse to the power of the number, (y*10+x)**(x*10+y), must end in the reverse, y*10+x, so ((y*10+x)**(x*10+y))%100==y*10+x.

Putting it all together we get a list of answers:

[x*10+y for x,y in [(a/10,a%10) for a in range(100)] if x!=y and ((x*10+y)**(x*10+y))%100==(x*10+y) and ((y*10+x)**(y*10+x))%100==(y*10+x) and ((y*10+x)**(x*10+y))%100==(y*10+x)]

Easy :)

PS: The answers are 16, 61 and 57