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Aug 06, 2023

Did you solve it? Are you smart enough for Mensa?

The solutions to today’s puzzles Today I set you these three problems by Mensa-approved author Barry R Clarke. Here they are again with solutions. 1. Switched on Three switches control three light

The solutions to today’s puzzles

Today I set you these three problems by Mensa-approved author Barry R Clarke. Here they are again with solutions.

1. Switched on

Three switches control three light bulbs, such that each switch controls only one bulb, and each bulb is controlled by only one switch. Only one of the following statements is true.

Switch 1: ‘Controls bulb B’.

Switch 2: ‘Controls bulb A or C’.

Switch 3: ‘Controls bulb A or B’.

Can you match the switches to the bulbs?

Solution: 1: C. 2: B. 3: A.

Only the last statement is true. If the first statement is true then the other two are false. This allows switch 1 to connect to B and switch 2 to B which is invalid. If the second statement is true then switch 2 controls A or C. Also the first and third are false so that switch 1 controls A or C, and switch 3 controls bulb C. Bulb B cannot be lit which is invalid. Finally, if the third statement is true then switch 3 controls A or B. The first two statements are false, so switch 1 controls A or C, and switch 2 controls B. So switch 3 controls A and switch 1 controls C.

2. Distance learning

Every afternoon, Jogger Jane runs from her home (left) to the school (right). Each of the four straight roads is 1km long and each of the four curved ones is 1.5km. She always runs more than 3km, and in doing so, she never passes along the same road twice. Not all roads are necessarily used in a single run, she can pass by her home, and once she reaches the school her run ends.

How many different routes can she choose from? (Hint: it’s more than 10.)

Solution: 16 routes.

3. Musical chairs

Six chairs numbered 1 to 6 are arranged sequentially in a circle for a game of musical chairs. When the music stops, six bottoms park themselves on six chairs, each chair being occupied by one person only. When seated, the players face inwards and the person whose birthday it is manages to sit in chair 1. The positions in the circle are as follows.

(1) Malcolm, who does not have the birthday, sits immediately to the right of Sally, who is not opposite the birthday person.

(2) Jennifer does not sit next to Uri.

(3) Nat is the first to sit down.

(4) Victor sits two places to the right of Jennifer.

(5) Uri sits at least two places from the birthday person.

Whose birthday is it?

Solution: Jennifer.

From (1), Malcolm sits immediately to Sally’s right, and from (4), Victor sits two places to the right of Jennifer. Viewing clockwise, this allows MSV_ J _ or MS_V_J. Considering (2), this allows only MSUVNJ. Using (5) to identify the birthday person, it is J, M or N. Condition (1) rules out M who does not have a birthday, and also eliminates N who is opposite Sally. So Jennifer has the birthday in chair 1, Malcolm is in 2, Sally is in 3, Uri is in 4, Victor is in 5, and Nat is in 6.

Thanks to Barry R Clark for today’s puzzles. They are from his brilliant book, Mathematical Conundrums, which came out last week.

I hope you enjoyed today’s puzzles. I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.