I’ve written an article for the Aperiodical entitled “Ask a mathematician: where should we live?”.
My partner and I are trying to buy a house. We both work in different places, and neither of us enjoys commuting. How could we decide where to live?
Thank you for your intriguing and entirely imaginary letter. The short and not terribly useful answer would be…
Want to know? Read the rest of it there.
In the first percentages for sceptics post, I showed that, if you are given a percentage, you can work out the minimum number of people to whom you would have to pose a yes-or-no question to be able to get that percentage. Ideally, I hope to add to your scepticism of percentages that are unaccompanied by the number of respondents. It’s easy to be suspicious of nice, round percentages like 10%, 20%, 50% etc., but in fact all but 14 of the whole number percentages can come from polls with 20 or fewer people.
The aim of this post is to take this approach to the next level. After a quick quiz, I’ll go through two examples, the second where I reverse-engineer a pie chart is the cleaner of the two. Don’t get hung up on any of the particulars of the numbers, especially in the dating example, what they are isn’t important, it’s more the fact that we can get them: most of the post functions as a demonstration of the principle.
Warm-up puzzle: A special case
In some survey 22% of people answered “yes”, 79% answered “no” (both to zero decimal places). Each person interviewed chose exactly one of the two options. What is the least number of people that could have been interviewed to get this result? Answer at the end of this post. It’s an on-topic mathematical question, not involving any silly tricks.
Let’s take a horrible press release reported as news by the Daily Mail, (commented on by the Neurobonkers blog) under the succinct headline: The dating rule book is being rewritten with one in four single girls dating three men at a time and a third happy to propose.
Given the trivial nature of the survey, alarm bells should be ringing; and the fact that it is “according to the study by restaurant chain T.G.I. Friday’s” means, like their food, this ‘research’ might best be taken with a pinch of salt.
[If you’re looking for the partial volume equations of a horizontal oil-tank, this article by Dan Jones has a nice write-up. Of course, you could just get someone to make a dipstick for you. My article only gives an overview of the problem, and how to approach it, from the perspective of a pure mathematician.]
This is the second post about real-life enquiries sent to my maths department, this one from a local engineer. Again, here’s the relevant section of the e-mail, partly to illustrate that, what might not at first seem an engaging problem for a pure mathematician, can turn out to be.
I need to calculate the capacity of a cylindrical tank laid horizontally with domed ends, in order to make an accurate dipstick. I would need to do the calculation several times in order to obtain capacities at varying levels.
This post is about how I went about finding a solution to this problem, and checking how good a solution it was. So now, if you happen to need a dipstick that fits these requirements (or just of a simple cylindrical tank), you can have one precision engineered to order. To continue the barge theme from the previous enquiry, you can also get a dipstick made for your barge’s diesel tank.
I’ve been learning a bit about Golomb rulers recently: a ruler which has so few markings that if you can use it to measure some whole number length, then you can only measure it in one way. I first read about them on the monthly AMS feature column, about their applications inside and outside of maths (to codes, radar, sonar and suchlike), and then watched an excellent TED talk using one particularly useful two-dimensional generalisation (a Costas array) to create a piece of piano music so dissonant that no time-step or jump in pitch between any pair of notes (not necessarily adjacent) is the same.
A perfect Golomb ruler with markings at 0,1,4 and 6, can measure any whole length from 1 to 6, but each in only one way.
I started to wonder about whether Golomb rulers had anything to do with a real-life problem I’ve previously written about, that someone who owned a barge had wanted answered. He asked about how to cut ropes into different lengths so you can knot them together in combinations and get a large variety of new lengths. I had decided the link was merely thematic, until someone else asked me whether they were the same, and prompted me to have a closer look. It turns out the two are somewhat linked, and what’s more, the link can be viewed as a silly little piece of mathemagic!
I was recently travelling on a budget flight with a friend, with no assigned seating. Walking up to the queue, we wondered whether we would be able to sit together. As people joined behind us, my friend, who whole-heartedly detests maths in any disguise, pointed out that we certainly could. To be so full-up to mean that we couldn’t, each row of three would need at least two people sitting in it. Since we could see from the queue that we had more than a third of the passengers still left to board after us, then as long as we weren’t trampled in a boarding stampede, we could certainly sit together.
This was actually a subconscious application of the pigeonhole principle, one of the most intuitive theorems that mathematicians use. It states that if you have objects to put in boxes then one of those boxes must contain at least two of the objects.
Or if you put pigeons in pigeonholes (or envelopes in pigeon-holes) then, if , at least one of the holes has at least two pigeons. So far, the type of maths you wouldn’t be shocked to see explained on television by a primary-coloured puppet.
Betting is often a good way of settling arguments: “I bet you so much that my sports team will beat yours”, can quickly lead to a resolution. But some of the most heated arguments are between people who completely agree with each other. So what happens when you want to gamble with someone else, but both of you want to bet on the same outcome?
You could bet on a particular score, or the timing of an event. But how about if the options are limited: is the correct direction to turn left or right? Or if you have identical views: the game will end in a nil-nil draw, or this party will win the election?
Here, the bet can go ahead as long as one of you is more sure than the other that they’re correct. Continue reading
Let’s suppose you see a newspaper article or an advert on television that claims “73% of women reported healthier looking hair” or “88% of cats prefer Meonards to other leading feline food brands”, but they don’t give the number of respondents to their survey (perhaps you also have to suppose it happened in the past—the Advertising Standards Agency in Britain seems to have clamped down on this behaviour). What is the minimum number of people they could have asked?
Clearly, we can get any integer percentage by asking 100 people (or pets with strong brand preferences), and to get, say, 75% we only need to have asked four (with three responding positively). But if we assume that they rounded to zero decimal places in the standard manner, the number of respondents may be far fewer than expected (or at least fewer than I expected). Continue reading