# My Mathematical Magic Show: Part 9

This mathematical trick was not part of my Pi Day magic show but probably should have been. I first read about this trick in one of Martin Gardner‘s books when I was a teenager, and it’s amazing how impressive this appears when performed. I particularly enjoy stumping my students with this trick, inviting them to figure out how on earth I pull it off.

Here’s a video of the trick, courtesy of Numberphile:

Summarizing, there’s a way of quickly determining $x$ given the value of $x^5$ if $x$ is a positive integer less than 100:

• The ones digit of $x$ will be the ones digit of $x^5$.
• The tens digit of $x$ can be obtained by listening to how big $x^5$ is. This requires a bit of memorization (and I agree with the above video that the hardest ones to quickly determine in a magic show are the ones less than $40^5$ and the ones that are slightly larger than a billion):
• 10: At least 10,000.
• 20: At least 3 million.
• 30: At least 24 million.
• 40: At least 100 million.
• 50: At least 300 million.
• 60: At least 750 million.
• 70: At least 1.6 billion.
• 80: At least 3.2 billion.
• 90: At least 5.9 billion.