A Flea’s Journey

Image: Wikimedia Commons

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.

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The Magic Total

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?

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Home Again

Image: Wikimedia Commons

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.

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“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.

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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.

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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?

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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.

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