Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart Page B

Your task is to divide the water into two parts, each of 4 litres, by pouring water from one jug into another. You are not allowed to estimate quantities by eye, so you can only stop pouring when one of the jugs involved becomes either full or empty.

    Divide the water into two equal parts.
    Answer on page 299

Alexander’s Horned Sphere
    If you draw a closed curve in the plane which doesn’t cross itself, then it seems pretty obvious that it must divide the plane into two regions: one inside the curve, the other outside it. But mathematical curves can be very wiggly, and it turns out that
this obvious statement is difficult to prove. Camille Jordan gave an attempted proof, more than 80 pages long, in a textbook published in several volumes between 1882 and 1887, but it turned out to be incomplete. Oswald Veblen found the first correct proof of this ‘Jordan curve theorem’ in 1905. In 2005, a team of mathematicians developed a proof suitable for computer verification - and verified it. The proof was 6,500 lines long.

    A closed curve, with the inside shaded.
    A subtler topological feature of such a closed curve is that the regions inside and outside the curve are topologically equivalent to the regions inside and outside an ordinary circle. This too may seem obvious, but, remarkably, the corresponding statement in three dimensions, which seems equally obvious, is actually false. That is: there is a surface in space, topologically equivalent to an ordinary sphere, whose inside is topologically equivalent to the inside of an ordinary sphere, but whose outside is not topologically equivalent to the outside of an ordinary sphere! Such a surface was discovered by James Waddell Alexander in 1924, and is called Alexander’s horned sphere. It is like a sphere that has sprouted a pair of horns, which divide repeatedly and intertwine.

    Alexander’s horned sphere.

The Sacred Principle of Mat
    The daredevil adventurer and treasure-hunter Colorado Smith, who is not like a real archaeologist at all, ducked a passing shower of blazing war-arrows to check the crude sketch-map scribbled in his father’s battered notebook.
    ‘The holy sanctuary of Pheedme-Pheedme the goddess of eating and sleeping,’ he read, ‘is formed from 64 identical square cushions stuffed with ostrich-down, arranged in an 8×8 array. The five sacred avatars of Pheedme-Pheedme, represented in overstuffed fabric, are to be placed on the cushions so that they “watch over” every other cushion: that is, every other cushion must be in line with one occupied by an avatar. This line can be horizontal, vertical or diagonal, relative to the array, where “diagonal” means “sloping at 45° ”.’
    ‘Look out!’ shrieked his sidekick Brunnhilde, taking cover beneath a large stone altar.
    ‘I wouldn’t do that if I were you,’ said Smith, and hauled her out a split second before the supporting slabs exploded in puffs of dust and the 10-tonne altar stone crashed to the ground. ‘Now, Dad’s notebook says something about the principles of—uh - Mat?’
    ‘Ma’at was the Egyptian concept of justice and rightful place,’ Brunnhilde pointed out. ‘But this temple is Burmalayan.’

    ‘True. Can’t be ma’at . . . No, it’s definitely the Principle of Mat. Apparently the goddess reclines on a mat, surrounded by her sacred avatars. We have to leave a space for the holy reclining mat, which is square. Hmm . . . maybe this would do.’

    Is this how to lay out the five sacred avatars and Pheedme-Pheedme’s reclining mat?
    ‘That seems suspiciously easy,’ said Brunnhilde. ‘What else do we have to do?’
    Smith quietly removed a deadly kamikaze-scorpion from her hair, hoping she wouldn’t notice. ‘Uh, we have to arrange the avatars to leave the largest possible space for a square mat. Bearing in mind that they must watch over every cushion. I doubt we can do better than my picture.’
    ‘Those ancient priests were sneaky, though,’ said