A and B are playing a simple game. Between them are nine tiles numbered 1 to 9. They take tiles alternately from the pile, and the first to collect three tiles that sum to 15 wins the game. Does the first player have a winning strategy?
A conduit carries 50 identical wires under a river, but their ends have not been labeled — you don’t know which ends on the west bank correspond to which on the east bank. To identify them, you can tie together the wires in pairs on the west bank, then row across the river and test the wires on the east bank to discover which pairs close a circuit and are thus connected.
Testing wires is easy, but rowing is hard. How can you plan the work to minimize your trips across the river?
Lori has an icky problem: Worms keep crawling onto her bed. She knows that worms can’t swim, so she puts each leg of the bed into a pail of water, but now the worms crawl up the walls of the room and drop onto her bed from the ceiling. She suspends a large canopy over the bed, but worms drop from the ceiling onto the canopy, creep over its edge to the underside, crawl over the bed, and drop.
Desperate, Lori installs a water-filled gutter around the perimeter of the canopy, but the worms drop from the ceiling onto the outer edge of the gutter, then crawl beneath. (The worms are very determined.) What can Lori do?
By W. Timbrell Pierce. White to mate in two moves.
Tim Krabbé calls this “one of the funniest chess problems I ever saw.” Its composer, M. Kirtley, won first prize with it in a Problemist tourney in 1986.
It’s a selfmate in 8, which means that White must force Black to checkmate him 8 moves, despite Black’s best efforts to avoid doing so.
The solution is a single line — all of Black’s moves are forced:
1. Nb1+ Kb3 2. Qd1+ Rc2 3. Bc1 axb6 4. Ra1 b5 5. Rh1 bxc4 6. Ke1 c3 7. Ng1 f3 8. Bf1 f2#
All of White’s pieces have returned to their starting squares!
A problem from the U.S.S.R. mathematical olympiad:
You’re given 13 gears. Each weighs an integral number of grams. Any 12 of them can be placed on a pan balance, with 6 in each pan, so that the scale is in equilibrium. Prove that all the gears must be of equal weight.
A puzzle by Pierre Berloquin:
In my house are a number of rooms. (A hall separated from the rest of the house by one or more doors counts as a room.) Each room has an even number of doors, including doors that lead outside. Is the total number of outside doors even or odd?
From The Booke of Meery Riddles, 1629:
A soldier that to Black-heath-field went,
Prayed an astronomer of his judgment,
Which wrote these words to him plainly,–
Thou shalt goe thither well and safely
And from thence come home alive againe
Never at that field shalt thou be slaine.
The soldier was slaine there at that field,
And yet the astronomer his promise held.
By Alexander Yarosh. The position above was reached in a legal game, except that one piece has been knocked off the board. What was it?
A problem from the 2004 Moscow Mathematical Olympiad:
An arithmetic progression consists of integers. The sum of its first n terms is a power of two. Prove that n is also a power of two.