A flea sits on one vertex of a regular tetrahedron. He hops continually from one vertex to another, resting for a minute between hops and choosing vertices without bias. Prove that, counting the first hop, we’d expect him to return to his starting point after four hops.

# Puzzles

# The Magic Total

Each of the 36 numbers in this table is the sum of the numbers at the head of its column and at the left of its row. For example, 3 = 2 + 1 and 13 = 5 + 8. The six bold numbers have been chosen so that each of them falls in a different row and a different column. The underlined numbers were chosen in the same way. But each of these two sextets produces the same total: 16 + 6 + 5 + 14 + 8 + 8 = 8 + 10 + 7 + 8 + 10 + 14 = 57. In fact, *any* six numbers chosen in this way will produce the total 57. Why is this?

# Home Again

On a regular 8 × 8 chessboard, a wandering knight can visit each square once and then return to his starting square. Show that he *can’t* do this on an *m* × *n* board if *m* and *n* are both odd.

# Black and White

By Francis Healey. White to mate in two moves.

# Black and White

This problem, by Charles Pelle, won third prize among the Union des Problémistes de France in 1946. How can White mate in two moves?

# Simple Enough

I think this appeared originally in Ed Southall’s Twitter feed. What fraction of the square is shaded?

# “A Chess Packing Problem”

In 2006 Martin Gardner asked: Can you arrange the 16 non-pawn pieces in a standard chess set on a 5 × 5 board so that no piece attacks a piece of the opposite color? As in a conventional game, the two bishops of each color must stand on squares of opposite colors.

# The Dark Half

A puzzle from the 1997 Ukrainian Mathematical Olympiad:

Cells of some rectangular board are coloured as chessboard cells. In each cell an integer is written. It is known that the sum of the numbers in each row is even and the sum of numbers in each column is even. Prove that the sum of all numbers in the black cells is even.

# Black and White

By K. Makovsky. White to mate in two moves.

In *The Two-Move Chess Problem* (1890), Benjamin Glover Laws calls the first move here “ideal” and “splendid.” “[I]t is not always a composer’s good fortune to strike a vein which is susceptible of such an excellent opening move as is illustrated in this problem.” What is it?

# Decomposition

This verse is known as “Lord Macaulay’s Last Riddle.” Lord Macaulay was Thomas Babington Macaulay (1800-1859), though his authorship of the riddle is uncertain. What’s the answer?

Let us look at it quite closely,

‘Tis a very ugly word,

And one that makes one shudder

Whenever it is heard.

It mayn’t be very wicked;

It must be always bad,

And speaks of sin and suffering

Enough to make one mad.

They say it is a compound word,

And that is very true;

And then they decompose it,

Which, of course, they’re free to do.

If, of the dozen letters

We take off the first three,

We have the nine remaining

As sad as they can be;

For, though it seems to make it less,

In fact it makes it more,

For it takes the brute creation in,

Which was left out before.

Let’s see if we can mend it —

It’s possible we may,

If only we divide it

In some new-fashioned way.

Instead of three and nine, then,

Let’s make it four and eight;

You’ll say it makes no difference,

At least not very great;

But only see the consequence!

That’s all that need be done

To change this mass of sadness

To unmitigated fun.

It clears off swords and pistols

Revolvers, bowie-knives,

And all the horrid weapons

By which men lose their lives;

It wakens holier voices —

And now joyfully is heard

The native sound of gladness

Compressed into one word!

Yes! Four and eight, my friends!

Let that be yours and mine,

Though all the hosts of demons

Rejoice in three and nine.