You and I play a game. I ask you to write down two numbers (positive integers like 2, 5, 34 and so on), one below the other as if you’re going to add them (which you will, shortly). Any numbers will work, but it might be best to start with small ones because of the arithmetic that’s ahead.
Done? Within a few seconds in which I furrow my brow and scratch my (non-existent) beard, I scribble something on a piece of paper, fold it and put it in your shirt pocket.
Now we really get going. I ask you to add the two numbers to produce a third, writing it below them. Add the second and third to produce a fourth. Add the third and fourth to produce a fifth… carry on like this until you’ve written ten numbers down, all in one long column.
Add the ten numbers. In the few seconds it takes for you to do this, I chew on a nail and twiddle my thumbs. When you have a total, I suggest that you yank out that piece of paper in your shirt pocket. You yank it out. You unfold it. You read out what’s on it. Voilà! It’s your total!
How does this work? (It really does).
Scroll down for the solution
The Weekly Puzzle Solution
This has to do with what are widely known as Fibonacci numbers — though some mathematicians, like Manjul Bhargava, believe they should be called Hemachandra numbers. That’s because the polymath Acharya Hemachandra described them in the 12th century, some 50 years before Leonardo Bonacci, known as Fibonacci, did.
If you start with 0 and 1 and do the same operations (each new number being the sum of the two preceding it), what you get is the Hemachandra (or Fibonacci) sequence.
This trick relies on one of the properties of Hemachandra numbers: Whatever two numbers you start with, the sum of the first ten in the sequence is the sum of 55 times the first and 88 times the second. So this is what I calculate in my head and write on that paper.
Now I don’t really know my 55- and 88-times tables. Instead, I calculate the sum of 5 times the first and 8 times the second — most of us know those multiplication tables fairly well and can do them in our heads at least for small numbers. But even if not, this number will actually be the 7th you write down. So if I’m not confident about multiplying by 5 and 8 and adding, I can also wait till you write your 7th number and use that.
Whether I’ve calculated or taken your 7th, I now multiply this number by 11 — which is again relatively easy to do mentally.
(For example, take 53 times 11. I write 5 and 3 with a small gap in between, add the two to get 8, which I put in the gap: answer, 583. If the addition produces a two-digit answer, put the units digit in the gap and add one to the number to the left of the gap. 68 x 11? 6+8 = 14. Leave the 4 in the gap, add 1 to 6, thus the answer is 748. The same idea can be extended to larger numbers.)
This number is what I write on my piece of paper.
For example, suppose you start with 2 and 7. I multiply 2 by 5, that’s 10. 7×8 = 56. 10+56 = 66. Now what’s 66 x 11? Put down 6 and 6 with a gap in between. Add 6+6 = 12. Leave the 2 in the gap, add 1 to 6, thus the answer is 726. That’s my prediction and that will be the sum of the ten numbers you produce.
Check: 2, 7, 9, 16, 25, 41, 66 (the 7th number), 107, 173, 280. Total 726.
Now you go do it.