In a shooting match, Andryusha, Volodya, and Borya each fired 6 shots, and each totaled 71 points.

Andryusha’s first 2 shots earned him 22 points, and Volodya’s first shot earned 3 points.

Who hit the bullseye?

George Sicherman, of Sicherman dice fame, has designed a puzzle to welcome the new year: Print out the PDF, cut out the letters MMXXVI (the Roman numerals for 2026), and fit them into the tray provided so that no pieces with the same shape touch one another.

Sicherman’s website has a whole range of puzzles, most of them geometric.

(Thanks, William.)

Does the sequence of squares contain an infinite arithmetic subsequence?

A problem from the October 1964 issue of Eureka, the journal of the Cambridge University Mathematical Society:

My friend tosses two coins and covers them with his hand. ‘Is there at least one “tail”?’ I ask. He affirms this (a).

Just then he accidentally knocks one of them to the floor (b). On finding the dropped coin under the table, we discover it to be a ‘tail’ (c).

‘That is all right,’ he says, ‘because it was a “tail” to start with.’ (d).

At each point (a), (b), (c) and (d) of this episode I calculated what, to the best of my knowledge, was the probability that both coins showed ‘tails’ at the time. What were these probabilities?

By W.F. Wills. White to mate in two moves.

In a letter to Maud Standen dated Dec. 18, 1877, Lewis Carroll included a puzzle:

[M]y ‘Anagrammatic Sonnet’ will be new to you. Each line has 4 feet, and each foot is an anagram, i. e., the letters of it can be re-arranged so as to make one word. Thus there are 24 anagrams, which will occupy your leisure moments for some time, I hope. Remember, I don’t limit myself to substantives, as some do. I should consider ‘we dishwished’ a fair anagram.

As to the war, try elm. I tried.
The wig cast in, I went to ride.
‘Ring? Yes.’ We rang. ‘Let’s rap.’ We don’t.
‘O shew her wit!’ As yet she won’t.
Saw eel in Rome. Dry one: he’s wet.
I am dry. O forge! Th’rogue! Why a net?

For example, the first foot in the first line, “As to,” can be rearranged to spell OATS. Carroll left no solution, but he did add a parting riddle to which we have the answer:

“To these you may add ‘abcdefgi,’ which makes a compound word — as good a word as ‘summer-house.'” What is it?

A gigantic tire, with a radius of 100 miles, is rolling down Broadway at 60 mph. One driver fails to notice the tire’s approach until its descending surface is just touching the roof of her car, 6 feet above the road. If she leaves the car immediately and can shrink to within 2 feet of the road’s surface, how long does she have to crawl out of the tire’s path?

The Xmas Puzzles 2025 is now live — 13 fiendish puzzles and a “metapuzzle” that draws on their solutions. The competition will run until 20:00 GMT on January 18.

Quizmaster Tim Paulden has pledged just over £1000 in charitable donations as prizes. The top four entries will win a donation to a charity or good cause nominated by the solver: £200 for first place, £150 for second, £120 for third, and £90 for fourth. Those who solve the metapuzzle or score 50 percent or more will also win a donation.

Entry is free and open to all — participants can work alone or in teams of up to five people. Details are at the link above.

A problem from the October 1961 issue of Eureka, the journal of the Cambridge University Mathematical Society:

“The map below represents one-way street system of a certain university city, the direction in which travel is allowed being indicated by arrows. An undergraduate living at A wishes to cycle round the city, visiting each intersection just once, and returning to A. What route must he take?”