Did you solve it? Riddles inspired by card tricks

Cards
 Photograph: Graham Turner for the Guide for the Guardian

Chess Boxing

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?

Solution

The game will always be a draw. The number of black and white pieces is the same, so if two white pieces are taken from the box, the box will have two more black pieces than white ones, so there must be a turn later in the game in which two black pieces are taken, levelling out the scores. If the score is 4-2, at least 8 white have been taken and 4 black, meaning that the box has 4 more black in it than white, since when one of each colour is taken the colour surplus does not change. If the box has 4 more black pieces than white pieces, this guarantees two points for the hustler, meaning the final score will be 4-4, or a higher scoring draw.


Tunnel Traffic

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 one, 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?

Solution:

None.

Imagine one lane is BWBWBW… and the other is WBWBWB… If the first car through the tunnel is a B, then car behind is either the second car in the first lane or the first car in the second lane, which is always a W. These two cars will exit together and do not match in colour.

The third car through the tunnel is either a W or a B. If it was a W, then this means that two lanes of remaining cars BOTH have a B at the front. (Sketch the order of the lanes down and cross them out as they go through the tunnel and you will see the pattern emerge). And if both have a B in the front, the fourth car must be a B. So, again, when the third and fourth cars exit together they will not match in colour. If the third car is a B, then the two lanes of cars must both have a W at the front, and so on…

The process of the two lines of traffic merging into one is the same as merging two sets of cards into one. The order of the cards within each set do not change relative to each other, and this produces the nice effect that when you count them out in twos from one end of the pack, you always get one of each 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?

Solution

100 per cent. Lets call the five sets of twins in age ascending order 1A and 1B, 2A and 2B, 3A and 3B, 4A and 4B and 5A and 5B.

So, Team A starts in this order 1A 2A 3A 4A 5A

Team B starts 5B 4B 3B 2B 1B

The last person replaces the first person six times. So either one team does it six times and the other zero, or five times and one time, or four and two, or three and three. You will see that in each case, the leaders of the teams are the same set of twins.

For example, let’s say Team A replaces its leader once. The order will then be 5A 1A 2A 3A 4A. And let’s say Team B replaces its leader five times. Which puts them back to the starting formation 5B 4B 3B 2B 1B. Twins 5A and 5B – who have the same age! – are the finishing captains for their teams.