Riddler Express: May 7, 2021

David Ding

May 7, 2021

Trio of Squares

Can you find three distinct numbers such that the second is the square of the first, the third is the square of the second, and the first is the square of the third? Assuming you can, what are three such numbers?

The first of the three numbers is the all non-1 complex solutions to the equation a7=1.

Explanation:

Without the 'distinct' clause, we can easily think of two numbers, 0 and 1, who fit the description. But alas, is life ever that easy? Let us have a triplet, (a,b,c), that represents the three numbers in the puzzle respectively. From the givens of the problem, we have:

b=a2c=b2=a4a=c2=b4=a8

Therefore, we are left to solve the following equation given that a0 and a1:

a8=aa7=1

Since a0, we can cancel out an a from both sides of the equation.

Adding complexity to the situation, no pun intended, we have 7 roots for the above polynomial equation. Of course, we must ignore the a=1 case as that would not make the other two numbers distinct from a. This leaves us with the following, for n=1,2,3,4,5,6:

a7=1a7=e2πnia=e2πni7

When n=7, we get back the real number 1 for e2πni7, and the cycle repeats beyond that every multiple of 7 for n. Also remember e(2k)πni=1 for all integer values of k.

So we have six triplets for (a,b,c) that satisfy our answer:

n a b=a2 c=b2 c2
1 e2πi7 e4πi7 e8πi7 e16πi7 = e(14+2)πi7 = e2πie2πi7=e2πi7=a
2 e4πi7 e8πi7 e16πi7 e32πi7 = e(28+4)πi7 = e4πie4πi7=e4πi7=a
3 e6πi7 e12πi7 e24πi7 e48πi7 = e(42+6)πi7 = e6πie6πi7=e6πi7=a
4 e8πi7 e16πi7 e32πi7 e64πi7 = e(56+8)πi7 = e8πie8πi7=e8πi7=a
5 e10πi7 e20πi7 e40πi7 e80πi7 = e(70+10)πi7 = e10πie10πi7=e10πi7=a
6 e12πi7 e24πi7 e48πi7 e96πi7 = e(84+12)πi7 = e12πie12πi7=e12πi7=a

This puzzle was "complex" after all!