Puzzle: Knights and Pages

[Update: Click here for the solution and analysis]

I am a regular reader of Richard Wiseman‘s blog, including the weekly Friday Puzzle (the answer to which is always given on the following Monday). The puzzles vary in quality, and often leave me unimpressed, but other times they are very interesting challenges.

About seven months ago, I sent Richard a puzzle that I thought would be well suited to the weekly challenge. At the time he told me he would definitely use it, so I mentioned it to the community of Friday Puzzle blog post commenters, saying that some day a puzzle I’d submitted would show up. However, one cannot wait forever — I’ve got a lengthy analysis for the maths geeks that’s been waiting in my drafts, and I can’t unveil that until I first unveil the puzzle.

So last month, as we passed the six-month anniversary of the day I submitted it, I realised it was time to make a decision. I’m revealing the puzzle on this blog, and I’ll discuss the solution in a seperate blog post next week (along with that analysis I mentioned).

The puzzle comes via the book The Chicken From Minsk And 99 Other Infuriating Brainteasers by Yuri B. Chernyak and Robert M. Rose (previously mentioned in my Bookmash post). It resembles the famous wolf/goat/cabbages puzzle, but is more interesting. Here it is:

  1. Many years ago, three knights waited to cross the river Neva. Each knight had his own page, so there were six people. The boat they had could only carry two. However, the knights were ferocious killers and the pages were terrified. In fact, it was certain that any one of the pages would die of heart failure if he were not protected at every instant from the other knights by the presence of his own master. Was there any way to cross everyone over the river without losing a page?
  2. Suppose the previous problem involved four pairs of knights and pages. Is there a solution?
  3. Reconsider the previous problem with four pairs of knights and pages, but with an island in the middle of the river. With the island, is there a solution?

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Fractal collection

I’m in the process of moving stuff from my old website to the blog, or to my new website, or to box.net, depending on where I think it belongs. Today I’d like to share my fractal collection.

The first one is generated by a formula that I stumbled upon in the nineties (when I originally created these images). Artistically, I see some sort of magical artifact, perhaps the knob from a wizard’s staff.

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Why Pi is Not Three

It’s fun to be whimsical, and one way to achieve that is to take a topic – however silly – and write about it, simply as an exercise to see where it leads and what angles can be illuminated on the journey. Now, various people have expressed the notion that mathematics would be a lot simpler if only pi were exactly three, and there are even stories in which the idea of pi being equal to three forms part of the plot (one modern example being Going Postal by Terry Pratchett). So it occurs to me that it might be fun to imagine a world in which pi really is equal to three, and to mentally explore some of the consequences.

A trivial way to render true the statement “pi equals three” is to invent a language in which the word for “three” is pronounced “pie“, but that is not only cheating, it also lacks subtlety. One way or another, cheating is a necessary part of the exercise, but we can at least be more subtle about it. Pi is the ratio of a circle’s diameter to its circumference, and if we want to imagine a world in which pi equals three we should at least preserve that definition.

A shape for which the perimeter really is exactly three times the “diameter” (defined here as the distance from corner to diametrically opposite corner) is of course the regular hexagon. A good way to show this is to divide the hexagon into six equilateral triangles. If we agree not to mess with the geometry of straight lines, then a universe in which pi equals three must be a universe in which the circumference of a circle is equal to the perimeter of the largest hexagon that can be inscribed within it. How can we make this be?


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Memories of Childcraft

In a shop recently, Mum found second-hand copies of the Childcraft series from World Book Encyclopedia, which were among my favourite books as a child. I’ve often remarked they’d make great gifts for relatives who are about the same age now, so with this in mind, she bought them.

The Childcraft series is still being published, but in a form so different from the books I had that they can scarcely be recognised (over the years they’ve gone through many different editions, with some changes to the titles at each revision). Reading the series was a very significant part of my childhood. I received them about one volume a month when I was about seven years old — as an extended birthday present — and their contribution to my quest for knowledge is incalculable.

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If you designed your own scientific calculator – nothing too fancy or revolutionary, there are no bonus points for breaking with tradition nor for seeing how many buttons you can fit on – where would you place the buttons to make the calculator most pleasing to your mind’s eye? There is elegance and intellectual beauty in a layout whereby buttons adjacent to each other implement similar functionality and there is a reason for every button being situated exactly where it is, but how exactly would you maximise the elegance of your calculator? In this post, I will consider my own answer.

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Prime squiggle

I have a computer program I once wrote which counts 2, 3, 4, 5, … all the way up to a specified termination point, and draws a squiggly line according to a simple mathematical rule.

Every time the counter passes a prime number, the line turns a 90 degree corner. Every time it passes a composite number, the line is extended one pixel forward. It goes up on four (the first composite), left on six (having passed the prime five), down on eight, nine and ten (having passed the prime seven), right on twelve, and so on.

[Update June 2007: The uncompiled Java program is available for download here. The “PrimeFinderMessGUI” Java application will display instructions on standard output.]

Here is a picture of what the line looks like after the program has counted to 100,000. Notice that up until about 11,000 (and there are over a thousand primes before 11,003) the overall trend is very distinctly vertical, rather like rising smoke. This means that every fourth interval between primes in that range tends to be larger than the rest, which is surprising. But wait! You ain’t seen nothing yet.

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