The Rice And Chessboard Story

By Michael Hartley

There's a famous legend about the origin of chess that goes like this. When the inventor of the game showed it to the emperor of India, the emperor was so impressed by the new game, that he said to the man

"Name your reward!"

The man responded,

"Oh emperor, my wishes are simple. I only wish for this. Give me one grain of rice for the first square of the chessboard, two grains for the next square, four for the next, eight for the next and so on for all 64 squares, with each square having double the number of grains as the square before."

The emperor agreed, amazed that the man had asked for such a small reward - or so he thought. After a week, his treasurer came back and informed him that the reward would add up to an astronomical sum, far greater than all the rice that could conceivably be produced in many many centuries!

How Much Rice? We are all like the emperor in some ways - we find it hard to grasp how fast functions like "doubling" makes numbers grow - these functions are called "exponential functions" and are actually found everywhere around us - in compound interest, inflation, moldy bread and populations of rabbits. Another puzzle involving exponential functions goes like this :

• A particular lake has water lilies growing on it. On the first day, there is one water lily. Each day, the number of water lilies doubles. After 30 days, the water lilies cover half the lake. How long before they also cover the other half of the lake, so the whole lake is full?

The answer is both obvious and very surprising - the lilies that took 30 days to cover half the lake take only one more day to cover the other half. They fill the lake on day 31.

I've prepared, here, some worksheets for kids to help them learn about this kind of amazing math. The worksheet goes with a story - a modern version, if you like, of the rice and chessboard legend. It goes like this.

You are offered a job, which lasts for 7 weeks. You get to choose your salary.

• Either, you get $100 for the first day, $200 for the second day, $300 for the third day. Each day you are paid $100 more than the day before.
• Or, you get 1 cent for the first day, 2 cents for the second day, 4 cents for the third day. Each day you are paid double what you were paid the day before.

Which do you choose?

Most people unfamiliar with this kind of dilemma will choose the first option. I've provided some printable worksheets to help people work out which is really best. The worksheet uses the currency symbols $ and c. If you are in the United Kingdom, you'll prefer the pounds and pence version. I've also made a version for those in the Eurozone.

You could use the worksheets like this.

• Present the story to a group of kids. Feel free to embellish it with lots of imaginative details!
• Ask the children to choose which "pay package" they prefer, either $100 for the first day, increasing by $100 per day, or 1 cent for the first day, doubling each day.
• Distribute the worksheets. Each worksheet has a grid for calculating the daily wage, and for recording the total earned for each week.
• Perhaps as part of their homework assignment, ask them to complete one row of the grid each week. On the next Monday, get them to report the weekly totals, and discuss. Maybe keep a week-by-week tally of how many kids think the first choice is better.
• The computations for the first pay package are simple enough. For the second, the numbers start to get large by the third week. You could allow the children to round off the answers to the nearest dollar, then hundred dollars - or, by weeks 4 and 5, thousand and million dollars.
• By week 4 it becomes clear that the second choice is better.

If you print the worksheets, you'll see that there are three worksheets, not just two. The third is useful if you also want to introduce the class to the Fibonacci numbers. Here, the deal is

• 1 cent on the first day, 2 cents on the second, then your salary on any day after that is your total salary on the previous two days.

To help you out, here are the correct weekly (and overall) totals for the three different schemes (without rounding).

• For the first choice, the person earns $2800 in week 1, then $7700, $12600, $17500, $22400, $27300 then $32200, for a grand total of $122,500
• For the doubling scheme, the person earns $1.27 in week 1, then $162.56, $20807.68, $2,663,383.04, $340,913,029.12, $43,636,867,727.36 then $5,585,519,069,102.08 for a total of $5,629,499,534,213.11.
• For the Fibonacci scheme, the totals are $0.53, $15.42, $447.71, $12,999.01, $377,419.00, $10,958,150.01 then $318,163,769.29, for a total of $329,512,800.97.

Enjoy!