Euclid, Pythagoras, Ptolemy, and Hypatia are playing a game where they all have to think of a number, and then cube that number 20 times. Hypatia doesn't want to cube large numbers, so she chooses the number 1. Euclid thinks the same thing and also chooses the number 1. However, Pythagoras and Ptolemy don't think ahead and Pythagoras chooses 2 and Ptolemy chooses -2. After they finish cubing their numbers (Pythagoras and Ptolemy take a while), all four players write their final numbers on a piece of paper. What is the sum of the numbers they wrote on the piece of paper?
Well, just sum them up like this:
(1^3)^20 + (1^3)^20 + (2^3)^20 + (-2^3)^20 =
What caught my eye in this question is the 2 and the -2.
Hypatia's and Euclid's numbers have a sum of 2. However, Plotemy cubes -2 twenty times. We can notice a pattern: \((-2)^3=-8,(-8)^3 = -512.\) and so on (when dealing with big numbers, look for patterns or try examples/smaller numbers). Answer this: Is there a cancellation that we can use?