Cribbage

From reader Derek Christie:

Each player in a game of cribbage has a hand of four cards. A single further card is turned up and serves as the fifth card in every player’s hand. Part of the game involves scoring your hand. You get points for any combination of cards that adds to 15, like 9 4 2; for two or more of any rank, like 3 3; or for any run of three or more, like Ace 2 3. The Jack, Queen, and King each score 10. Show that if a 5 has been turned up, every player must score some points.

	Select Click for Answer
Try and make a hand that doesn’t score. The turnup is a 5. This means you must choose 4 different additional cards from the group 1 2 3 4 6 7 8 9, and to avoid scoring, the resulting hand mustn’t include any pairs or runs or any combination of cards that adds to 10. Since you can’t have two cards adding to 10, you’ll need to choose one card from each of the pairs 9 1, 8 2, 7 3, and 6 4. Try 6 from 6 4. Now you have 5 6. You can’t have 7 (which would produce a run), so you must take the 3, giving 5 6 3. But this is a dead end: Now you must avoid both the 1 and the 9, since each of these produces a 15. So you can’t get anywhere with the 6 from the 6 4 pair. Try the other card from that pair, the 4. Now you have 4 5. You can’t have 3 (producing a run), so you must take the 7, giving 4 5 7. Now you can’t take 8 (since this would produce 15), so you must take the 2, giving 2 4 5 7. And now both the 1 and the 9 again give 15, another dead end. With both the 6 and the 4 excluded, there’s no way to assemble a hand of four cards that can remain scoreless with a turnup card of 5. So every hand must score something. (Thanks, Derek.)

Click for Answer

February 6, 2022February 5, 2022 | Puzzles

Black and White

huggins chess problem

By A.Z. Huggins. White to mate in two moves.

	Select Click for Answer
1. Ne4! and the queen must mate. (Knotty little puzzle!) From Chess Problems, 1872.

January 31, 2022January 30, 2022 | Puzzles

A Miserable Vacation

twine puzzle

Your eccentric uncle has dropped you into the middle of Twine Island, which is festooned with one continuous loop of twine. The twine never crosses itself, but it snakes everywhere, and the island is too hilly for you to see the whole layout at once. As a character-building exercise, your uncle offers you a million dollars if you can determine whether you’re inside the loop or outside. How can you do this?

	Select Click for Answer
Walk in any direction until you reach the shore, counting the number of times you cross the twine. If the final number is odd then your starting point was inside the loop; if not then you were outside. Related: Is it possible to draw a straight line that intersects every side of a 999-sided polygon? No. A straight line can’t intersect every side of any polygon with an odd number of sides. Each time the line crosses a side, one of the polygon’s vertices falls on one side of the line and another vertex falls on the other side. If we succeeded there would be an equal number of vertices on each side of the line, but an odd-sided polygon has an odd number of vertices. (That last one is by Polish mathematician Paul Vaderlind.)

Click for Answer

January 28, 2022January 27, 2022 | Puzzles

Footgear

City A contains 20,000 people. One percent of these have one foot only and wear one shoe apiece. Half of the remaining people go barefoot, wearing no shoes at all, and the rest wear two shoes apiece.

In City B, 20 percent of the residents have one foot only and wear one shoe apiece. Of the remainder, half go barefoot and half wear two shoes apiece.

If 20,000 shoes are worn in City B, what is its population?

	Select Click for Answer
20,000. Just as in City A, it really doesn’t matter what proportion of the population has one foot only. Because the rest of the population also averages one shoe per person, so does the town as a whole. In both cities, the number of residents equals the number of shoes. (A Puzzler from NPR’s Car Talk.)

January 21, 2022January 20, 2022 | Puzzles

Black and White

pierce chess problem

By James Pierce. White to mate in two moves.

	Select Click for Answer
1. e6! and White will mate with bishop or pawn. From Chess Problems, 1873.

January 10, 2022January 9, 2022 | Puzzles

A Simple Plan

https://commons.wikimedia.org/wiki/File:Cuboid_abcd.svg — Image: Wikimedia Commons

You have three identical bricks and a ruler. How can you determine the length of a brick’s interior diagonal without any calculation?

	Select Click for Answer
Stack two of the bricks and place the third alongside. Now you’ve identified the corners of an invisible fourth brick, whose diagonal you can measure directly with the ruler. (Bonus: If the bricks’ edges and face diagonals all have integer lengths, and you find that the interior diagonal is an integer as well, you’ll have solved an open problem in mathematics.) 01/03/2021 UPDATE: Reader Paul-Georg Becker points out that the task can be accomplished with two bricks and a ruler: (Thanks, Paul-Georg.)

Click for Answer

January 3, 2022January 3, 2022 | Puzzles · Science & Math

RSS Quiz

The Royal Statistical Society has released its Christmas quiz for 2021, a set of 11 puzzles that require general knowledge, logic, lateral thinking, and searching skills, but no specialist mathematical knowledge.

Anyone may enter, individually or in teams of up to five. The top entry will receive £150 in Wiley book vouchers, second place £50 in book vouchers, and the next three entries a puzzle book or board game. And everyone who achieves a score of 50 percent or higher will win a donation to their favorite charity or good cause.

Entries must be received by 21:00 GMT on January 31. See the quiz web page for the rules and some tips for budding solvers.

December 25, 2021 | Puzzles

A Christmas Puzzle

Mathematician Matthew Scroggs has released this year’s Christmas card for Chalkdust magazine. (Actually I’m late — it’s been out since December 4.)

Solving 14 puzzles will reveal a Christmas-themed picture in this diagram.

You can download the card as a PDF or complete it online.

December 22, 2021December 21, 2021 | Puzzles

Black and White

taylor chess problem

By J. Paul Taylor. White to mate in two moves.

	Select Click for Answer
1. Nc4! and the queen will mate. From Elementary Chess Problems, 1880.

December 20, 2021December 19, 2021 | Puzzles

Crowd Control

A knight’s tour is a series of moves by a chess knight such that it visits each square on the chessboard once. The example above is a “closed” tour because it ends on the square where it started.

This inspired a puzzle posed by Martin Gardner. If we filled a standard chessboard with knights, one on each square, could all 64 of them move simultaneously? The closed knight’s tour shows that they could — they form a long conga line, with each knight vacating a square for the knight behind it to occupy.

Gardner asks: Could the same feat be accomplished on a 5 × 5 chessboard?

	Select Click for Answer
The answer is no, and we can prove it by observing the board’s coloring. A 5 × 5 chessboard has 13 squares of one color and 12 of another. The nature of a knight’s move means that it always jumps from a square of one color to a square of another. So on a 5 × 5 chessboard, 13 knights would have to jump onto 12 squares, which is impossible. So the task can’t be done.

December 15, 2021December 14, 2021 | Puzzles