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.

# Puzzles

# 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.

# Tight Squeeze

Above: From Paris, 1927: a novelty car that can “sidle” into parking spaces.

Below: Someone was actually working on this in the 1950s (thanks, Martin):

A related puzzle from *The Chicken From Minsk*, Yuri B. Chernyak’s 1995 collection of math and physics problems: Why is it easier to parallel-park a (conventional) car by backing into the space rather than pulling in directly?

# A Narrative Alphametic

James Mayfield gives this clever expression in the August 2006 issue of *Word Ways*:

BEAR + RARE + ERE = RHYME

It’s an alphametic: If each letter is taken to represent a different digit, it encodes a valid equation. What’s the equation?

# Cutting Up

Is it possible to divide a circle into *n* parts of equal area using only a straightedge and compasses?

# Counter Play 2

Lee Sallows sent this clever puzzle, a followup to one we presented in 2010:

In the square shown above, any 3 counters in a straight line sum to 15.

Puzzle: Reposition the counters (again, one to each cell) to produce a new square again showing eight collinear triplets summing to 15, but with 1 now placed in a corner square.

# Black and White

A densely imaginative little problem by Joseph C.J. Wainwright. White to mate in two moves.

# “To My Pupil”

Lewis Carroll’s 1885 puzzle book *A Tangled Tale* bears an anonymous dedication:

Beloved Pupil! Tamed by thee, Addish-, Subtrac-, Multiplica-tion, Division, Fractions, Rule of Three, Attest thy deft manipulation! Then onward! Let the voice of Fame From Age to Age repeat thy story, Till thou has won thyself a name Exceeding even Euclid's glory!

He was thinking of Edith Rix, a child-friend who went on to study mathematics at Cambridge. She might have divined without asking that the dedication was intended for her. How?