Prove that if n is a positive integer greater than 1, then

is not an integer.

A problem from the 1996-1997 Estonian Mathematical Olympiad:

A square tabletop measures 3n × 3n. Each unit square is either red or blue. Each red square that doesn’t lie at the edge of the table has exactly five blue squares among its eight neighbors. Each blue square that doesn’t lie at the edge of the table has exactly four red squares among its eight neighbors. How many squares of each color make up the tabletop?

English chess enthusiast V.R. Parton invented this variant in 1953. The game is played on two boards, starting with the normal opening position on one board and a second board empty. The trick is that a piece that completes its move on one board “vanishes strangely off its board to appear suddenly on the other board, magically out of thin air!”

There are two stipulations: A move must be legal on the board on which it’s played, and the square to which the piece is transferred on the opposite board must be vacant. (This means that a piece can make a capture only on the board on which it stands.)

This makes things very confusing. Here’s a simple three-move game. White plays 1. e4, which means that his king pawn advances from e2, passes “through the looking glass,” and arrives at e4 on the second board. Black responds with 1. … d5, so his queen pawn arrives at d5 on the second board. White plays 2. Be2, meaning that his king’s bishop is transferred to e2 on the second board. And now Black makes the mistake of playing 2. … dxe4 — his pawn on d5 on the second board captures the white pawn there, which means the white pawn is removed from play and the black pawn finds itself on e4 on the first board. That’s a mistake, because White can play 3. Bb5:

The white bishop returns to the first board, checking the black king. Strangely, this is checkmate: Black can’t capture the attacking piece or move out of danger, and he can’t interpose a piece — any piece that he moves on the first board will find itself on the second, and there are no pieces on the second board that could move to c6 or d7.

Parton wrote, “One board being as a looking-glass to the other, the resulting play is a game which has a character as fantastic perhaps as Alice’s own game in Through the Looking-Glass.”

Here’s an Alice chess problem by Udo Marks. How can White mate in two moves?

A few weeks after his first confusing journey home from the train station, Smith again finishes work ahead of schedule and takes an early train home. This time he arrives at his suburban station half an hour early. Again, rather than wait for the chauffeur, he starts walking home. And as before, he meets his chauffeur on the road, who picks him up promptly and takes him home. How many minutes early do they reach the house this time?

Canadian puzzle designer Jeff Widderich invented this game in 2007. The goal is to place a digit 1-9 in each white cell so that each crossword-style “word” contains a straight, that is, a set of consecutive numbers in some order. For example, the top row of five digits might contain 62534, but not 91548.

One other constraint: Each full row or column must contain no repeated digits. That means, for example, that each of the two long vertical “words” will contain all nine digits. The digits in black cells count toward this constraint — the 9 in the black cell near the center means that no 9 appears elsewhere in its row or column. Can you complete the rest of the diagram?

Sam Loyd’s 1903 Eighth Book of Tan explores the world of tangrams, the pastime of constructing specified shapes from a given set of seven pieces:

The book includes a few “paradoxes,” two of which I’ve mentioned here before. But here’s another:

“The seventh and eighth figures represent the mysterious square, built with seven pieces; then with one corner clipped off, and still the same seven pieces employed.”

The book includes no solution. The square on the left is just the regular “block” formation above. But if anyone has discovered how Loyd produced a “clipped” square using the same seven pieces, I haven’t been able to find it.

12/22/2020 UPDATE: Reader Alon Shaham came up with this solution:

Here the seven pieces are used to make both figures, rather than each figure separately. Arguably Loyd’s description is artfully ambiguous. (Thanks also to reader Andrew Davison.)

By Johan Scheel. White to mate in two moves.

From a collection in Frank Mittler’s Little Book of Word Tricks (1958):

1. Pray tell me, listener, if you can,
Who is that highly-favored man
Who, though he marries many a wife,
May still stay single all his life?

2. I sit in fire, but not in the flame;
I follow the master, but not the dame;
I’m found in the church, but not in the steeple;
I belong to the monarch, but not the people.

3. Its light was mellow, soft and lazy;
One foot broke off — and it went crazy!

4. What is found in the very center of both America and Australia?

5. What divides by uniting and unites by dividing?

6. Why is a popular crooner like a doctor in an asylum?

Mike Keith published this remarkable invention in the Journal of Recreational Mathematics in 1975. Suppose a chess game goes:

P-K4   P-K4 
B-B4   P-R4 
Q-B3   P-R4 
QxP 

This is checkmate, so White says “I win!” Now if the game score is written out in one column, including White’s exclamation:

P-K4
P-K4
B-B4
P-R4
Q-B3
P-R4
 QxP
----
IWIN

… we get a solvable alphametic — replace each letter (and the symbols x and -) with a unique digit and you get a valid sum. (The digits already shown count as numbers, and those numbers also remain available to replace letters.)

This example isn’t quite perfect — the two moves P-R4 are ambiguous, as is the final pawn capture. Keith’s Alphametics Page has an even better specimen, a pretty variation from a real 1912 game by Siegbert Tarrasch. It’s 13 moves long and forms a perfect alphametic with a unique solution.