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 given the value of if is a positive integer less than 100:

• The ones digit of  will be the ones digit of .
• The tens digit of can be obtained by listening to how big 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 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.