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Grains of rice on chessboard problem

My grandson, Jack, now 12, called this week and told me about something that he had learnt at school: it was the old problem of grains of rice on a chessboard. I was pleased to learn that it was his history teacher, Mr Pearce, not his maths teacher, who had told him about this nice piece of mathematics and caught his imagination!

The story is about someone requesting 1 grain of rice on the first square of a chessboard, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number each time. The total number of grains of rice involved turns out to be an astounding number; much, much more than the number of grains of rice produced in the whole world in one year.

The mathematical series involved here is: 1 + 2¹ + 2² + 2³ + 2⁴ + ... + 2⁶³. The total of this is actually just 1 less than 2⁶⁴.

At a quick estimate, I made it about 1.8 × 10¹⁹. That's 18 followed by 18 zeros. Jack (and his Dad) seemed a bit disappointed that the number wasn't bigger than this!

How might we get a sense of the size of a number like this? I suggested we think about how long it would take to count this number of grains of rice, counting at the rate of 2 per second. A quick calculation on a calculator gave us the result: counting day and night, never stopping, it would take a person 300,000,000,000 years! Three hundred, thousand, million years! OK? It's a big number, yes?

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