In a regular pentagon, all diagonals are drawn, as shown. Label each vertex of the pentagon and each intersection of the diagonals with the number 1. Now: In one step you can change the signs of all the numbers on a side or on a diagonal. Is it possible, by a sequence of such steps, to convert all the labels in the diagram to -1?
The names of 13 Jane Austen characters are hidden in the following lines as anagrams of complete consecutive words. For example, “was ill” yields WALLIS. (The names to be found are women’s first names and men’s surnames, as in Austen.) In most cases the anagrams are hidden in two words, but twice they’re in three, once in four, and once in a single word. What are they?
The other day when I was ill
And not a soul I knew came nigh,
Jane Austen was my daily fare —
I rather liked to be laid by.
Each line or page enthralls me quite,
I there can let no man deride;
I may be ill as a wight can be,
But, Jane with me, am satisfied.
In bed my ease is nil, yet I’ll
Be lying therein at any rate
Content. With Jane to chortle at
How can I rail at Fate?
A problem from the qualification round of the 2004/2005 Swedish Mathematical Contest, via the April 2009 issue of Crux Mathematicorum:
The cities A, B, C, D, and E are connected by straight roads (more than two cities may lie on the same road). The distance from A to B, and from C to D, is 3 km. The distance from B to D is 1 km, from A to C it is 5 km, from D to E it is 4 km, and finally, from A to E it is 8 km. Determine the distance from C to E.
In Two-Move Chess Problems (1986), Robert Moore calls this a “very lightweight jewel.” White to mate in two moves.
Each of three houses must receive water, gas, and electricity. Is it possible to arrange the connections so that no lines cross?
No, it ain’t. Remove one house and draw connections to the other two:
This divides the plane into three regions, here colored red, yellow, and blue. Placing the third house into any of these regions denies it access to the correspondingly colored utility. So the task is impossible.
Pleasingly, the task can be accomplished on a Möbius strip:
And a torus can accommodate up to four houses and four utilities:
(By Wikimedia user CMG Lee.)
08/31/2025 UPDATE: Reader Guy Bolton King points out that Mathsgear sells a mug embossed with the puzzle. The joke here is that this makes the puzzle solvable — like the torus, the mug is of topological genus 1, “a blob with 1 hole in it,” so it admits the same solution.
And reader Shane Speck writes, “My sneaky solution … has always hinged on the fact that in reality, houses don’t all have separate pipes, and popping on a shared water pipe instantly reduces the problem to the status of incredibly trivial”:
“The problem doesn’t, after all, say you can’t do that… :)”
(Thanks, Guy and Shane.)
What’s the final digit in the product of all the odd numbers from 1 to 99?
If an angle of one degree is viewed through a lens with 4x magnification, how big will the angle appear to be?
Francesco Segala (1535-1592) made his name as a sculptor in Padua, but he’s remembered as a father of the picture maze.
Make your way from the traveler’s cup to the exit at bottom center.
