Ladies and gentlemen,
Today, three magic tricks. I mean three riddles.
A hustler and a chess grandmaster are playing the following game. All the pieces from a chess set are in a box. The two participants take turns reaching into the box without looking, picking out two pieces at a time and placing them onto the table. If the two pieces are white, the grandmaster gets a point. If the two pieces are black, the hustler gets a point. If the colours don’t match, no one gets a point. They will take turns removing pairs until the box is empty. If, exactly halfway through the game, the score is 4-2 in favour of the grandmaster, who will ultimately win the game and by how many points?
A boulder falls onto a mountain road, temporarily blocking access to a tunnel. A few drivers get out of their cars to help move the boulder. After they successfully clear the entrance to the tunnel, they notice some unusual traffic has accumulated on the two-lane road behind them. The traffic jam consists of eighteen white cars and eighteen black cars. The first car in the left lane is white and the first car in the right lane is black. The colours of the cars behind them in each lane alternate perfectly.
When passing through the narrow tunnel, the two lanes of traffic merge into trone, and when they come out the other side the single lane branches to two toll booths. Assuming that the first car through the tunnel is a black car and the last car through is a white car, and that the cars reach the toll booth in pairs – meaning that the first two cars exit together, then the next two exit together, and so on – what is the greatest possible number of exiting pairs that will match in colour?
Family Bike Race
At the Froome family reunion, five sets of twins from five different generations decide to have a bike race, with two teams and one sibling from each set of twins on each team. Each team sets off with all five cyclists in a line behind their “captain”. At any point in the race, the last person in the line can cycle to the front of the line to become the new captain. All five team members must cross the finish line to complete the race.
Both teams line up in age order but Team A starts the race with a little kid as their captain and Team B starts the race with an elderly woman as their captain. During the course of the race, the captains change six times between the two teams. When the teams cross the finish line, what are the odds that their captains will be the same age?
You may have noticed that each of these riddles involve two different kinds of the same thing: black chess pieces and white chess pieces, black and white cars, members of Team A and members of Team B. That’s because the mathematical principles behind the riddles are taken from card tricks, in which cards are either black or red. With a bit of thought, you can probably figure out just how the tricks work. Compared to the usual level in this column, today’s puzzles are relatively easy, but the mathematical patterns they reveal are interesting, satisfying and surprising, which is why magicians have plundered them.
The puzzles were devised by this column’s favourite cardsharp, Adam Rubin, who works with many internationally-known magicians. He is also Director of Puzzles and Games for the site Art of Play, on which you can buy beautiful playing cards.
I’ll be back with the answers at 5pm UK time.
NO SPOILERS. Please converse about cards.