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.