Tag Archives: puzzle

Arithm’ and clues in Puzzlebomb

I had my second puzzle in the January 2015 online puzzle sheet Puzzlebomb, entitled “Arithm’ and clues“. It’s a series of cryptarithm puzzles with the same letters denoting the same numbers throughout, but with none of the words given: instead they’re clued. It’s like a crossword but with arithmetic replacing the grid.

Go on and give it a try before reading on, as there are some tiny hints ahead. Continue reading

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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.

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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.

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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!

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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?

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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.