Think of a number…a maths party trick

I am writing this at our house in Rosthwaite, the Lake District. The neighbouring village is Stonethwaite, the wettest village in England. The combination of (I) location (ii) it being a Bank Holiday means it is pouring down outside, so we are keeping warm in the cottage.

In the last few weeks’ I have been explaining for probably the last ever time how dividend grossing up calculations work: for the last time, because tax credits are being abolished from 6th April. So twenty plus years of explaining ‘the company pays a net dividend, which is grossed up for notional tax paid by the company; then the total is taxed; then you deduct the notional credit the company hasn’t paid but you pretended it had’…has left clients (I) confused; (ii) happy they didn’t follow a career in tax.

Merely explaining the calculations is enough to result in most clients wishing they hadn’t asked for an explanation, but it has been even worse for a client whose dividends passed through a discretionary trust before received by individuals. Then you have to also explain the second layer of grossing up.

Explaining it all to said client, I remarked it was like the children’s party maths game. Probably an oxymoron, or possibly showing how unfortunate my children have been in the choice of party daddy.

Here it is (with some embellishments)

Think of a number

Think of a number between 1 and 9; suggest they don’t choose 1, because it is less interesting. Suggest that the braver they are, choose a larger number, but warn it is only for the brave, since the multiplication is harder.

Multiply the number by 9. Tell them they should now have a two digit number.

Swap the digits round. Explain that if their number is say 28 (eagle eyed readers note: I know it won’t be) ask them to swap the digits round, making 82.

Add to the two digits. Eagle eyed readers note: I know the ‘swap the digits round’ stage has no affect on the sum, but it helps confuse or prolong.

Deduct 4.

Then ask them to work out the letter based on A=1, B=2, and so on, and, quick as a flash!, think of an animal beginning with that letter.

The answer will be Elephant.

I tried it on my client, over lunch, and, of course, it worked: and then on my 18 year old daughter who…thought it was fun (amazing outcome), so, emboldened, tried on my primary school teacher wife, who also liked it, and asked me to write it down so she could try it on her pupils.

Now back to the wet weekend. That has given me time to work out why it works.

Below is the proof. Not entirely Euclidean rigorous, and it relies on the observation that the ten’s digit is always one less than the number which is multiplied by 9 (so 3*9 starts with a 2). I imagine Mod arithmetic would give a stronger proof, but hopefully my scribbled version is sufficient. The nub is that the sum of the digits is always 9, taking 4 off 9 gives 5, which is letter E, and Elephant is the only animal beginning with E that anyone can think of. I think.

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