Chess Boards, Exponential Growth and the Ice Bucket Challenge

Yesterday I was finally nominated for the Ice Bucket Challenge, I had actually thought this inevitable for the reasons in this post, but then it all kind of passed by. As it happens, it was my Daughter Aimee’s fault in the end. :-).

So I thought I would use the opportunity to bore people about the arithmetic behind it, in the end, I suspect that wasn’t too coherent since I was being watered at the time by my older Daughter Aimee, while my younger Daughter Matilda generally screamed at both of us.

So here is what I was trying to say. I like to talk to my students about a new version of an old problem. There’s an old story about a chess game where the winner will take away a certain amount of rice. The amount is calculated by having one grain on the first square, two on the next, four on the next, and so on, doubling all the way to the 64th square. My modified version is to consider coins, a UK 10 pence piece to be precise, piling up on the squares as we go along. The question is, how high is the pile of coins on the last square? I encourage you to guess, approximately what sort of size that is.

We double each time, so the number of coins on the first eight squares are 1, 2, 4, 8, 16, 32, 64, 128.
Another way to write that is
20, 21, 22, 23, 24, 25, 26, 27.

So, we are going along 64 squares, and start at 0, so the number of coins on the last square will be
263. This is a big number, but just how big? Do we change our guess on this information?

Now 263 = 9,223,000,000,000,000,000 = 9.223 x 1018
and a ten pence piece is 1.85 mm thick, or 0.00185 m, so when we multiply these we get

17,063,000,000,000,000 m = 1.7063 x1016 m.

Wow. That seems like a lot. Just how big is that number as a distance?

To put it in perspective…

  • by the 19th square, the coins are higher than the radio mast on the Empire State building;
  • by the 29th square, the pile of coins would have reached the Moon;
  • by the 48th square, the pile of coins would have reached the Sun;
  • by the 63rd square, one from the end, the coins are a light year high.

This is an example of the staggering power of exponential growth, and just how unintuitive it is.

So what has this got to do with the Ice Bucket Challenge? Well suppose you start with a single individual, who then nominates three people, each of whom nominate three people. Now on the Chess Board we have 1, 3, 9, 27 and so on. This is, again, exponential growth with powers of three rather than two (actually somewhat higher growth).

Mathematically these are often called Geometric Progressions or just G.P.s for short. These are sequences of the type

a, ar, ar2, ar3, ar4, …

There is a formula that can be derived (it’s not hard, the derivation is on the above Wikipedia link) for the Sum of the first n terms. (In Mathematics, contrary to popular opinion, a Sum specifically means the result of an addition process).

S_n = \frac{a(1-r^n)}{(1-r)}

In the above case we can see that a = 1 because that’s the first number in 1, 3, 9, … and r = 3 because that is the number we are multiplying by each time. So here

S_n = \frac{a(1-r^n)}{(1-r)} = \frac{(1-3^n)}{-2}

In other words, every “generation” each person nominates a further three people, so the number of people added each generation ramps up exponentially, and the total number involved increased rapidly too.

Generation New People Cumulative Total
1 1 1
2 3 4
3 9 13
4 27 40
5 81 121
6 243 364
7 729 1,093
8 2,187 3,280
9 6,561 9,841
10 19,683 29,524
11 59,049 88,573
12 177,147 265,720
13 531,441 797,161
14 1,594,323 2,391,484
15 4,782,969 7,174,453
16 14,348,907 21,523,360
17 43,046,721 64,570,081
18 129,140,163 193,710,244
19 387,420,489 581,130,733
20 1,162,261,467 1,743,392,200
21 3,486,784,401 5,230,176,601
22 10,460,353,203 15,690,529,804

So you can see that after just 22 steps in the nomination process, you have literally more people than you have on the planet. Pyramid selling schemes become untenable for this reason too… you quickly run out of suckers on any given land mass.

So if this really ran its course unimpeded (and I know that the original challenge was money or dowsing, with a requirement to undertake the challenge in 48 hours (some say 24) then the whole planet would have done the challenge in at most six weeks. If each person had used say 4 litres of water then at the time of writing the 7,259,289,122 people alive today would have collectively used 29 billion litres of water. Since an Olympic swimming pool has 2.5 million litres of water, that represents over 11,614 swimming pools filled with water. While it is easy to criticise those who have been uneasy about the water wastage here, to be honest they have a good point. Also one organisation would have billions of charity income rather than it being spread.

Clearly this has not happened[citation needed] so people have run out on enthusiasm here and there. Anyway, have fun, and give some money to something you love, if you want, when you want to.

I am donating to

Since people I care about have been troubled by both, and

since it can’t be denied I live in extraordinary luxury to be able to waste perfectly good water on this kind of thing.

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