Here are seven pennies, all heads up. In a single move you can turn over any four of them. By repeatedly making such moves, can you eventually turn all seven pennies tails up?

# Puzzles

# Black and White

By Francis Healey. White to mate in two moves.

# Opposites Exact

Prove that, at any given moment, there are two points on the equator that are diametrically opposed yet have the same temperature.

# Black and White

By J.H. Finlinson. White to mate in two moves.

# Injured List

Another puzzle from Henry Dudeney:

“It is a glorious game!” an enthusiast was heard to exclaim. “At the close of last season, of the footballers of my acquaintance, four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms.” Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with?

# The Dwarfs Problem

From the 1977 all-Soviet-Union Mathematical Olympiad:

Seven dwarfs are sitting at a round table. Each has a cup, and some cups contain milk. Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them. After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk. How much milk does each cup contain, if there were 42 ounces of milk altogether?

# Tiling Task

We’ve removed two squares from this 7×8 grid, so that it numbers 54 squares. Can it be covered orthogonally with tiles like the one at right, each of which covers exactly three squares?

# The Two Poles

A puzzle by Henry Dudeney:

A man planted two poles upright in level ground. One pole was six and a half feet and the other seven feet seven inches above ground. From the top of each pole he tied a string to the bottom of the other — just where it entered the ground. Now, what height above the ground was the point where the two strings crossed one another? The hasty reader will perhaps say, “You have forgotten to tell us how far the poles were apart.” But that point is of no consequence whatever, as it does not affect the answer!

# Black and White

By R.F. Bennett. White to mate in two moves.

# Out of Rank

A puzzle from John Beasley’s *The Mathematics of Games* (2006): Black has just moved. What is the smallest number of moves that can have been played in this game?