# Category Archives: Puzzle

## Solution to In Their Prime

Did you have a go at my In their Prime puzzle in Puzzlebomb yet? How about the other puzzles?

If your answer to these questions is “No”, then please turn to  page 13091204281 of the internet to have a go at the July Puzzlebomb.

If the answer is “Yes”, you can check the July solutions, including the numeric solution to In their Prime. But how did you find the solution? You may have successfully used trial and error, but that’s not usually very enlightening; you may have programmed a computer to do the dirty work for you (the puzzle was hand designed, but I did check the answer was unique with a bit of code). As a responsible puzzle-setter, I came up with the following possible proof of the solution. I’d be interested to hear from anyone who had a different method.

Filed under Puzzle

## In Their Prime in Puzzlebomb

I’ve contributed a one-off family tree puzzle called “In Their Prime” to the July issue of the excellent monthly puzzle publication Puzzlebomb. It’s about an extended family who die when they reach a semiprime (composite of two primes) age determined by their parents’ ‘prime number’ genes. My aim was to design a puzzle with a “How do I get started?” flavour which probably would be lost if there was a sequel without a sufficiently interesting twist.

If you haven’t already read them, there are two posts on the Aperiodical by Paul Taylor explaining the computer creation of and maths behind two puzzles that have also featured in Puzzlebomb:

• The extremely unique fractalphile “Hilbert’s Space-Filling Crossword” (only one non-trivial such puzzle exists).
• The more abundant “Spelling Bees” which have appeared several times in past issues (May, June and the aforementioned July issue), involve finding Hamiltonian paths that spell out a pair of words or phrases.

Also in the May issue, I especially liked “Word Split” a pentomino-based colouring word search (but I cheated by not breaking out the crayons). All issues can be found on the Puzzlebomb section of the Aperiodical.

Filed under Puzzle

## 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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## 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

## Algebraic Linesweeper

[If you want to dive straight into Algebraic Linesweeper, here's a printable PDF with rules.]

A friend of mine, Jak Marshall, created a pen-and-paper puzzle game called Linesweeper. Here are the rules:

• Draw a single, closed, continuous loop in the empty cells of the grid which never crosses itself or branches.
• The number clue in a cell indicates how many of the 8 adjacent cells are part of the loop (as in the classic computer game Minesweeper); for instance, a ’0′ means that none of the adjacent cells are part of the loop.
• The loop may not enter a cell with a number.
• The loop may run horizontally or vertically (not diagonally) between centres of adjacent cells (that is, parallel to the grid lines).
• The loop does not need to pass through all the unoccupied cells.
• Each puzzle should have a unique solution.
Here’s an example, with its solution immediately below:
You can find this example and at least 20 more online Java puzzles at the German puzzle site janko.at. [Edit: Andrea Sabbatini has included Linesweeper ("the looping minesweeper") into her 56 Logic Game Time Killers puzzle pack free for the iPhone and iPad. Linewsweeper works well on a touch screen.] Cross+A has even included Linesweeper in their commercial puzzle solver.

I think it’s fun and relaxing, and more satisfying to progress and finally connect the loop than fill in the final cells of a Sudoko puzzle. You can also design puzzles that use a bit of simple graph theory: if there is an area that your line visits with only three possible routes in and out, it can only use an even number (two) of these (the loop is in some sense a Eulerian circuit).

Algebraic Linesweeper

I’d like to introduce a variant of Linesweeper that I created. Instead of giving the number clues for a puzzle in a straightforward manner, I only give some letters that stand in for them. To be more specific:
• Each letter corresponds to a unique number between 0 and 8.
• Within a puzzle, different letters must correspond to different numbers.
• Each puzzle should still have a unique solution!
I’ve made a handy one-page printable pdf file of six Algebraic Linesweeper puzzles, with the rules. The result, at first glance seems impossible. To convince you it is, and check that you’ve understood, here’re two tiny practice puzzles:
The “Elementary” puzzle illustrates the point I was making about Eulerian circuits above: you can view the puzzle as a more discrete version of the following diagram inspired by the Bridges of Königsberg.
Here’s the six actual puzzles from the above pdf in an approximate order of difficulty. Obviously, the choice of letters is slightly arbitrary, so I thought I would choose them based on some theme. If I can spell out words, like below, then it gives a convenient title to identify each puzzle:

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## Blitz Krieger

I thought the following story would make a good question to add flavour to some first-year undergraduate Number Theory courses. It’s relevant, interesting, well-documented, and shows how a certain amount of scientific literacy can be extremely useful when evaluating bogus claims. Here’s an extract from a 1938 TIME Magazine article (££)

In 1937 A.D., a German-Jewish mathematician named Samuel Isaac Krieger, who was taking a mineral bath near Buffalo, N. Y., suddenly leaped out, rushed naked into the adjoining room, began to scribble figures. He thought he had discovered something too: a solution to the equation given in Fermat’s last theorem.

Krieger claimed to have found some integer $n > 2$ such that the equation $1324^n + 731^n =1961^n$ held, but refused to reveal the value of $n$.  A journalist quickly proved this to be false. Can you see how?

Obviously, Andrew Wiles only proved Fermat’s Last Theorem (that there are no integers $a,b,c > 0$ such that $a^n + b^n = c^n$ for $n > 2$) to be true in full generality after 1994, so he couldn’t use that. While it was known at the time that the theorem was true for exponents including $n=3,4,5,7$, the journalist only used elementary mathematics and did not need to use any specific knowledge about the theorem.

Solution after the break. Also, how not to solve FLT, and more on Krieger.