## Dice and Dissection: a puzzle

This is a puzzle, first appearing in Martin Gardner’s column in 1978, with a new way of thinking about the solution. Before the puzzle, though, a cultural diversion.

Rolling it in

In the modern classic board game The Settlers of Catan, it’s very important to know, when you roll a pair of dice, the frequency with which each number occurs. Resources are given players only if they have a settlement adjacent to those tiles whose number is rolled. If you build your settlements next to a tile labelled 2 or 12, it will, on average, only be productive once every 36 rolls. Tiles labelled 6 or 8 will produce resources five times in 36 rolls. It’s so fundamental to the gameplay that the relative frequencies are visualised as dots on the pieces: sixes and eights are so important, they are marked in red (rolling a seven does something different).

Puzzle

The well-known puzzle is:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

If I didn’t require you to use a positive number of spots on each face: then dice labelled {0, 1, 2, 3, 4, 5} and {2, 3, 4, 5, 6, 7} would work as a pair. If you allow negative numbers, there’s infinitely many solutions!

Filed under Accessible, Puzzle

## London MathsJam February Recap

Here’s a rough summary of February’s London MathsJam. There seemed to be some loose themes, but sadly no pancakes (it was on Shrove Tuesday). Peter Rowlett briefly visited, but left before most people turned up and the action started (the official MathsJam start time of 7pm is also the start of off-peak travel on the tube, so people tend to arrive later). We had about ten people in all, down from thirty-odd at January’s, when we took over the whole upstairs of the pub. There’s been a good mix of people in various walks of life, though most (but not all) had (or are doing) maths or computer science degrees: but everyone likes puzzles and games. This isn’t a full round-up: people sometimes split off into smaller groups, so it’s hard to keep track of everything, and there’s lots of chit-chat along the way that I haven’t documented.

♥ Someone autobiographically wondered what the chances of having two fire alarms in a day is.

♣ People were concerned when I brought out this noughts and crosses tiling puzzle (Think Tac Toe from this puzzle series), worrying at first it might be a physical copy of game itself:

The only solution anyone found wasn’t one of the four given on the back of the box:

Though it seems an unlikely to occur in a real game, it is a valid game position, so the solution is valid.

♣ Because noughts and crosses was universally unloved, we suggested replacements: Sim was explained, and 3D tic-tac-toe was played (on a 4×4×4 grid).

◊ Can you fit five rectangles together to form a square, where the rectangle side-lengths are each of the whole numbers 1 to 10? How many ways are there?

♦ Can you fit all twelve pentominoes, and an additional 2×2 square into: an 8×8 square; or into a 4×16 rectangle. We didn’t have time to try this one, or have any pentominoes handy.

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Filed under Puzzle

## Percentages for sceptics: part III

I wanted to do some self-criticism of my previous two posts in this series:

1. You can calculate the minimum of responses from a single percentage by hand (no need for computer programmes or look-up tables).
2. I’ve made a very rough model to estimate how many people the program typically returns when fed six percentages (as I did several times here).
In between, I’ve collected some links to demonstrate how great continued fractions can be.

Handy calculations

There are many ways of writing real numbers (fractions and irrationals) apart from in decimal notation. You can represent them in binary, for instance $\pi = 11.001001000011\ldots$, or in other bases. These have their uses: there is a formula to calculate the $n$th binary digit of $\pi$ without calculating all the preceding digits.

For our purposes we will use continued fractions. People write $\pi$ as $[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, ...]$: this notation means that 3 is a good first approximation to $\pi$, the well-known $3+\frac{1}{7}=\frac{22}{7}$ is the closest you can be with any fraction $\frac{p}{q}$ with $q \leq 7$. Then $3+\frac{1}{ 7 + \frac{1}{15}}=\frac{333}{106}$ is the best with $q\leq 106$, and the fourth term

$3+\frac{1}{7+\frac{1}{15+\frac{1}{1}}}=\frac{335}{113}$

is a very good approximation, as the next number in the square brackets, 292, is very large (I’ll motivate this observation at the end of the section). The golden ratio is sometimes called the ‘most irrational’ number because it has a continued fraction expansion with all ones, so the sequence converges slowly.

Filed under Accessible, Applications

## Percentages for Sceptics: Part II

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.

Dating data

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.

Filed under Applications, Maths in Life, Puzzle

## Kill the Dragon: a solution

This is my solution to the “Kill the Dragon!” puzzle. Improvements, in both the bounds and formality of the argument, are definitely possible.

Filed under Puzzle

## Kill the Dragon puzzle

Kill the Dragon!

A hapless lost dragon has accidentally landed in a nearby lake. You, the kingdom’s sworn dragon-slayer, have set out on this foggy night to kill it. You are armed with your trusty trebuchet, which can catapult a fiery projectile to any location on the lake. When the projectile hits the water, it will explode in a lethal circle of Greek fire, killing everything within a radius of $r$ metres from the point of impact. Especially dragons.

The fire, however is short-lived, and is extinguished instantaneously. This means the dragon, who swims slowly at a constant speed of $v$ metres per minute, can safely doggy-paddle into a previously scorched area. You can launch one missile per minute.

It’s so foggy, that you can’t tell whether you’ve killed the dragon, which is too tired to leave the lake, and you can’t be bothered to fetch a boat to check. If the lake is a circle of radius $R$ metres, is it possible to aim your volleys strategically to be sure that you will eventually kill the dragon, no matter how it moves?

For which radius $R$ is it possible, and for which is it impossible? What about other shapes of lakes?

Filed under Puzzle

## Oil tanks and dipsticks

[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.