Maths for Reassurance

In this generally interesting period of time, when attempts to join the Schengen free movement area are inhibited by it being shut or whatever it is that’s happening, it is important to hold onto positive news and whatever fixed points can be found.

So on the positive side we can all be reassured that the panic-buying in the shops isn’t being done by Millenials (those born between about 1980 and 2000) because the shops are still well-endowed with avocados and phone chargers.

On the fixed side, a late-night walk recently had me thinking – as these things do, particularly when it’s dark – about the method of calculating the Difference of Two Squares. This is a mathematical concept where the difference of two square numbers equals the sum times the difference.

Or, put in practical terms so it’s more understandable, if you know that 1² is 1 then you can work out 2² by adding 1 to 2 (which equals 3) and then working out the difference between 1 and 2 (which is 1) and multiplying the results of these two sums by each other (3×1) and adding the answer (which is 3) to 1² and getting 4. And in this case it works, because 2² is 4.

For further practical demonstration, suppose it is a commonly-known fact that 37² is 1,369, but you want to know what 52² is without all the difficulty, faff and waste paper of doing 52×52. The sum of 37+52 is 89 and the difference is 15. 15 by 89 is 1,335, so add that to 1,369 to get 2,704. And the square root of 2,704 is 52.

However, even an infinite number of practical demonstrations will not show that this must always work. Similarly (to use what I understand might be a genuine scientific example for this point), a biologist in a park could demonstrate that the world contains a number of white swans but cannot use this one park (or indeed thirty parks that coincidentally have no Australian black swans) to prove that all swans are white. Only political polling samplers can do that, using a not wholly scientific concept called extrapolation.

To demonstrate this square idea properly we must reduce it to algebra, where the lower number being squared will be shown as x and the higher number as y:

x² + (x+y)(y-x) = y²

If done in practice this would be, for example, 2² + (2+11)(11-2) = 11².

We then open out the brackets by multiplying each term in one set of brackets by everything in the other bracket, thus:

x² + xy – x² – xy + y² = y²

Or, when done in practice, 2² + (2×11) – 2² – (2×11) + 11² = 11².

As x² – x² (or 4-4) is 0 and xy – xy (or 22-22) is 0 we can cancel those things out to get:

y² = y²

Or, in practice, 11² = 11², which works out as 121 = 121.

Isn’t that satisfying?

And it’s very nice to know that, whatever may happen in the world, and however much you may distrust the Government, and regardless of the fates of the airlines, 121 tubs of discounted ice-cream that everyone else has mysteriously forgotten to panic-buy will always be equal to 121 tubs of ice-cream.

Meanwhile, here is a picture taken on a late-night walk:

(Not the walk that inspired this blogpost. That one was a bit different, as it was a new moon, and up on a plateau, and nowhere near the sea.)

Leasehold Mathematics

The Beeb has decided to spend the day explaining why you don’t want to buy a leasehold property.

Recently I picked up a book on the history of housing, which remarked (almost in so many words) that the point of leasehold was that you owned the land, got a house built on it and leased it out for the life expectancy of the house. After the lease was up you got a tatty old house back which you knocked down (or rather your descendants knocked down) and replaced with a new, up-to-date house, maintaining the quality of the area. (And as all the houses would be on the same leasehold they could be knocked down simultaneously.)

This worked fine until someone came up with the idea of lease extensions, with the result that all these cheaply built Georgian leasehold houses remain highly desirable properties on short leases. Despite houses now having a higher life expectancy (it having been realised that a cheap Georgian 5-storey terrace does not fall down after 80 years) leaseholds remain a peculiarly popular way of dealing with perfectly ordinary houses which at first glance look like freehold.

And, of course, they are handy for freehold houses turned into several flats (as not all the flats can own the freehold and the landlord would probably rather it if none of them did).

Which brings us to this week’s sob story. This chap has bought a leasehold flat.

Its lease lasts for 190 years and the lease cost doubles every ten years (and is, of course, on top of the £150k that he paid for the right to live in the flat and pay this lovely lease).

It starts at £250 per annum.

Readers of Murderous Maths (or indeed anyone with a reasonable grasp of arithmetic) will rapidly realise this may be going to get messy (this chap obviously wasn’t brought up on Murderous Maths and Asterix the Gaul) – the total rent he will have paid over the lease is seen here totted up by decade end:

1. £2,500 (some people pay this in monthly rent);
2. £7,500 (a slightly excessive annual rent for a flat – but remember this is a total divided over 20 years);
3. £17,500 (about what I paid in total rent over four years for a flat, but divided over 30 years);
4. £37,500;
5. £77,500;
6. £157,500 (bit more than my mortgage, spread out over 60 years instead of 25);
7. £317,500;
8. £637,500;
9. £1,277,500 (which would currently buy the landlord a technology-stuffed self-powered railway carriage – after 90 years of patient totting-up);
10. £2,557,500;
11. £5,117,500;
12. £10,237,500;
13. £20,477,500;
14. £40,957,500;
15. £81,917,500;
16. £163,837,500;
17. £327,677,500;
18. £655,357,500;
19. £1,310,717,500 (which will then allow the landlord to spend the total takings by paying the premium for a moderately prosperous intercity rail franchise, and makes an average annual rent of about £6.9million on a flat).

The formula for working out how much will have been paid in total by the end of each decade is something to the effect of:

(2,500 x 2^) – 2,500 

^ represents multiplying the 2 by the power of the decade number (there should really be a little elevated x instead, but that takes mucking around with HTML superscript codes). So for the 7th decade you multiply 2,500 by 2 to the power of 7 (or 2 x 2 x 2 x 2 x 2 x 2 x 2) (and get 320,000) then subtract 2,500 (which is 317,500).

Put the 2,500 back on again and divide by 2 to work out the amount to be paid over that decade – thusly:

(2,500 x 2^) / 2

or indeed

2,500 x 2^-1

Which for the 7th decade is £160,000, or £16,000 per annum (£250 x 2 x 2 x 2 x 2 x 2 x 2 – ignoring the last x 2 in the upper of these two formulas because it’s cancelled out by dividing by 2 – and aren’t formulas neat?)

This formula is very handy for working out how much you’ll have to pay in ground rent for a flat on a leasehold arrangement like this one if you stay there for ten years.

On the other hand, this sort of contract – with its annual rent in the last decade of £250 x 2^18, or £65,536,000 per annum – is a rather nifty bet against inflation.

(If somewhat overkill. £250 in 1826 would officially now be worth (2016 prices) £22,756.58.)

This is a house, set in several hundred acres of landscaped countryside and possibly actually worth £6.9million (though not per annum).

Light Maths

The next post on this blog will be rather heavy, so here’s something lighter in a similarly mathematical vein to get people in the mood:

(x+8)²+(x+7)²+(x+6)²=(x(x+5)²)+(x+4)²(x+3)²+(x+2)²(x+1)²-(x+1)(x-1)+x

Find “x”.

(Assuming that my maths is good enough for it to work…)