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.

§2.1 Funny numbers

Simple arithmetic using every-day sorts of numbers produces straightforward answers. That is such a commonplace observation that it needs no examples. But the advent of the electronic calculator has made it easy to explore simple arithmetic with not-quite-so-every-day decimals. Surprises soon appear.

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.

§2.2 Car costs

Let's suppose you want to know how much it costs to run your car. You'd know how to set about it, probably along these lines:

• You would have a good idea as to how many miles the car does per litre of fuel, and the cost of each litre. So divide the one by the other and you have the fuel cost per mile.
• Longer-term, you could probably record or estimate the average cost of a routine service which would include oil and perhaps the occasional new tyre. Divide that figure by the 6,000 or whatever the service mileage is and you have the maintenance cost per mile.
• But there are also a number of nasty lumps -
  • There's the annual licence. No problem: you know what that cost.
  • The case is similar with the insurance premium - annual, fixed cost regardless of mileage covered. There might be the membership charge for a breakdown organisation too.
  • But then there's that horrible thing called depreciation. Even if we don't call it that, we do know that sooner or later we'll need to replace the car with a new (or newer) one: we ought to be setting aside some savings so that the cash is there when the time comes. It's really a bit of future-predicting so it can't be exact: a guess must do.
    Put those three together and you get a figure to add to the day-by-day, month-by-month costs.
• Purists might want to add some other odds and ends - perhaps quite significant odds and ends in some cases. Like the cost of garaging. It can be expensive to rent a garage. Or if you have your own as part of your home then a proportion of any mortgage ought really to be charged to the car. And mentioning the mortgage raises the subject of interest payments. Was the car bought on hire purchase or with a loan perhaps? If so there will be interest to pay, but if you bought it outright then you could, otherwise, have had that money invested to produce interest on the credit side for you. So either way the car ought to bear some form of interest charge, even if it has to be pretty much of a guess rather than anything very exact.

If you've been able to put figures in under all these headings you will have finished up with a total cost. Divide that by the total number of miles and there you are - total cost per mile! Let's suppose, for the sake of argument, that you have finished up with 60 pence per mile. So if you were thinking of offering a taxi service you'd need to charge a minimum of 61 pence per mile to make a profit. Obviously. But wrong!

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.

• If a friend wanted to come on a journey with you, what would be the extra cost to you? Virtually nothing: the reduction in miles per litre would increase the fuel used by an almost negligible amount which would be more than adequately covered at 1 penny per mile.
• Sharing the cost of the petrol is another basis on which journeys are sometimes shared. Just a few pence per mile covers it.
• But it's much less clear what it would mean to "share the costs equally". Would the friend be expected to contribute towards the interest charges and the garaging costs? The car depreciates anyway, regardless of how many miles it goes let alone how many people are in it for each mile. And the same goes for tax and insurance of course. It's more likely that fuel and maintenance costs were intended. Certainly it's more a problem of defining terms than of simple arithmetic.
• All those journeys were shared. But what about that possible taxi service? There the meaningful questions are not about cost per mile but of cost per EXTRA mile. Clearly all the annual charges can be left out of account completely since none of them increase with the extra mileage. What we need to do is to charge enough, over and above the fuel and maintenance costs, to contribute something towards those fixed annual costs. If charges are low enough sufficient customers should be attracted to produce an income which covers all the costs and leaves something over, at the end of the year, as profit.

The point of the discussion is not to learn about taxi finances: it is simply to realise that obvious answers are not necessarily the meaningful answers, even in matters of simple arithmetic. Business management experts embroider these ideas with more precise definitions, involving fixed costs and variable costs, gross margins and perhaps gross profits. First-blush intuition is not likely to be useful.

§2.3 Dice

When we go gambling with dice, as promised in the Introduction, we shall be entering a blind alley. There are logical (even if counter-intuitive) reasons for asking particular questions in management accounting: what we are going to look at next is just simply a mystery. It is most certainly counter-intuitive, but if any sensible questions can be found which have reasonable answers then I have yet to find them or hear of anyone who has.

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
B consistently beats C

- and you don't need to have a degree in logic to realise that -

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: