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.
Tag Archives: puzzle
Did you have a go at my In their Prime puzzle in Puzzlebomb yet? How about the other puzzles?
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.
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.
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 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 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 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 is it possible, and for which is it impossible? What about other shapes of lakes?
[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.
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).
- 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 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 such that the equation held, but refused to reveal the value of . 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 such that for ) 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 , 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.