What I want to do in this Chapter is to share with you an exploration of some cases in which it is sensible to question what simple arithmetic proves. Now I know some forms of figuring have a very bad press - that comment about "Lies, damned lies, and statistics" just for one. "You can prove anything with figures" may be true in the hands of an expert, but that doesn't include me.
We shall not need to go into any complicated or advanced mathematics: no calculus or trigonometry or even Cartesian co-ordinates. If you should want to check absolutely everything in the Chapter all you will need is a pocket calculator and three dice. Briefly, we shall have a look at some of the funny ways of numbers themselves, consider the profits of a possible taxi business, and then go gambling.
We can take any decimal between zero and one (other than exactly 0.5): we are going to multiply it by itself and by two and take one away. Then we're going to repeat the squaring and multiplying and taking one away - repeatedly. So for an example run the keys to press on an ordinary simple calculator might be . (decimal point) 6 5 4 3 2 1, then × (multiply) and = does the squaring, × 2 doubles, and − (minus) 1 = does the taking away. My calculator produces -0.1437282: nothing remarkable. The first repeat (or 'iteration') is done without clearing that number so it's just × = × 2 − 1 =. The second answer? -0.9586846. The next × = × 2 − 1 = takes it to 0.8381522. And so on, to 0.4049982 then -0.671953 and -0.0969584 and -0.9811982 and 0.9254998 and so on - and on - and on with no pattern whatsoever. Indeed, the series goes on for ever without ever having values of 1 or 0 or 0.5. There is no magic in the .654321 starting point: any other decimal, positive or negative, produces a similarly chaotic sequence.
A computer can speed things up, of course. So if a spreadsheet is preferred the starting figure can be put into cell A1. Then the formula for A2 is =A1 × A1 × 2 − 1. It is then a very simple matter to get the computer itself to copy that into every cell as far down column A as you like. (So cell A50, for example, will get =A49 × A49 × 2 − 1.) Whenever you change the start value in A1 the sequence zips down the column faster than thought. Better still perhaps, if the spreadsheet can display the results as a graph it is even easier to appreciate the chaotic irregularity of the results. It may even be possible to watch the (absence of) pattern as the start figure in A1 is changed.
At a very much more sophisticated level "chaos" is apparently a branch of mathematics nowadays. None of this has any great philosophical significance: I am not suggesting that the world is chaotic. Rather am I hoping to show that, just as seeing and hearing, tasting and feeling are not as simple as they seem to be, so 'simple' mathematics needs only a very modest extension to reach strange lands - to turn counter-intuitive.
But we can turn now to perhaps even simpler sums, certainly in a more down-to-earth context.
Surprisingly perhaps, there's no one single figure that can sensibly be taken as a cost per mile: the truest figure depends on what you want the information for. Let's take some examples.
I suppose by this time you are on your guard against my making some very questionable statements. But if I said that I can show you that something called "A" is better than another similar one called "B" which is in turn better than a third, "C", there would be nothing at all remarkable about it. There would just be a perfectly ordinary series A, B, C in decreasing order of quality. You know from the heading that we are going to be considering dice so the "better" must refer to being a better bet - to yielding higher scores when thrown at random. And they are perfectly ordinary dice with no bias nor weighting, and the throwing is indeed purely at random since we cannot put on any twists or prevent any spinning.
So OK then, we have got three dice and -
A consistently beats B
A must consistently beat C.Now what is actually going to happen, in accordance with purely mathematical rules, is that -
C consistently beats A.If you find that remarkable, read on: if you don't you may as well skip to Chapter Three. It is exactly on a par with the second floor ("A") being above the first floor ("B") and the first floor ("B") being above the ground floor ("C") AND the ground floor ("C") being above the second floor ("A")! It is illogical, unreasonable, incoherent - but not impossible, as we are about to see.
Please note that it is NOT on a par with the children's game in which stone blunts scissors, scissors cut paper, and paper wraps stone: in that case a different criterion in used for each pair, whereas in the dice paradox we are simply to use numerical scores in every case.
Dice always have six faces with six (normally different) numbers on them. But consider three special dice with these following numbers on their six faces:
(The reason for choosing B to have 3 on every face is simply to keep our later sums as simple as possible. I know you wouldn't go gambling with such a die. And I know we are supposed to be dealing with a game of chance when there is no chance of B showing anything but 3. But please bear with me for the time being - as I say, for the sake of simplicity: we shall have a look at the exactly similar case of dice with different numbers on their faces towards the end of §2.4.)
Each die has a dot-total of 18, such as 4+4+4+4+1+1. And since every face is equally likely to appear it is clear that no die can be any "better" or higher-scoring than any other. False! And what is worse, the impossible but true situation is that over any reasonably long series of throws:
A consistently beats B
Since the dice are not weighted in any way, each face of any die is equally likely (in the long run) to appear. So if we were to throw A 3600 times the chances are that the outcome would be something like -
600 of the first face with score 4So from the 3600 throws there will about -
2400 scores of 4We already know that B is going to score 3 every time it is thrown.
So when A (4 4 4 4 1 1) plays B (3 3 3 3 3 3) we logically and mathematically expect that:
A would win about 2400 times
Again, all the B scores are bound to be 3s: the expected 3600 scores for C will be around:
600 of the first score 5 so C winsso
B would win about 2400 times
It takes a bit longer to work out the chances with A (4 4 4 4 1 1) playing C (5 5 2 2 2 2): we no longer have the convenience of a die which always scores 3. But we can carry on at the very basic no-tricks-up-my-sleeve level: the method is still straightforward even if it is a bit more tedious. As before, 3600 throws of A are likely to produce something like -
600 of the first score 4While A scores 600 of its first "4"s C will score something like -
100 of the first score 5 so C wins- to give A 400 wins, and C 200 wins.
Exactly the same chances occur with A's other three "4"s, to give a total so far of A 1600 wins, and C 800 wins.
Combining the other two options, while A scores 1200 of its "1"s C is bound to win every time with its scores of "2" or "5". So all told -
A would win about 1600 times
Those would be normal six-sided dice. But the relationships would hold true in exactly the same way if it were possible to conceive of three-sided dice. Although impractical in reality, the idea serves to simplify the spreadsheet a great deal. So we'll adopt the simplified specification -
The dot totals happen to be eleven for Red and Blue but only ten for Green. The 6 could be changed to 7 but it would not make any difference to the relationships: feel free to use 7 in your own spreadsheet if you prefer. The important thing is that there are no equal numbers on different dice, so there can never be a tie. In a similar way, the number of 'throws' in each 'game' must be odd so that, again, no ties are possible. And the actual number of 'throws' must be large enough to give randomness a chance to work itself out but need not be huge: 41 will do.
Many spreadsheets share identical or similar codings so the following are likely to fit your set-up.
That has set up the first throws of the dice. Thanks to the magic of spreadsheets all these formulae can then be copied right down to and including row 43. So cell F43, for example, stores =IF(E43=1,6,2).
Now we need to find and record the winners of each throw. There will be three pairs to compare so we can set up the headings and then make the comparisons.
As before, G3, H3 and I3 can be copied and pasted right down to G43, H43 and I43. Then all that remains is to find out which member of each pair has 'won'. Since there have been 41 'throws' in each case the winner is whichever has scored more than 20. It will be user-friendly if the outcome can be shown in writing, preferably at the top where is it easily visible, perhaps along these lines -
The first 'game' will have been revealed as the spreadsheet was being built. After Saving it, any "Recalculate" (possibly F9) produces a whole new 'game'. Experiment!
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The next chapter concerns Paradoxes.
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