Can you logic your way to €1m?

1 Nov 2016

Image: paulista/Shutterstock

Logic is the foundation of our digital world. It’s also a handy mathematical device used for solving maths puzzles and brainteasers.

“Cara, you’re just one answer away from winning all of the prizes on the board, as well as €1m. All you have to do is get this next question right. Are you ready to play the final round of Mathemagic?”

Cara took a beat before answering. Not because she needed to – she had dispatched her former competitors, including recently retired super-detective George Boole, with ease – but because she understood the nature of TV gameshows. It was all about the suspense.

“Yes, Frank, I’m ready to play.”

Mathemagic host Frank L Rhine explained the final round for the audience at home.

“Here in front of us, we have seven identical envelopes, labelled with the numbers 1, 2, 3, 4, 5, 6 and 7. A token has been placed in one of the envelopes. Without touching the envelopes, or examining them in any other way, Cara must find that token in order to win her prizes.

“The only tools she can use to find the token are the clues written on this card. To add a little difficulty for Cara, two of those clues are false.”

Rhine handed Cara the card, from which she read the five clues.

  1. The main prize can be found in an envelope with an odd number label.
  2. The main prize can be found in one of the following envelopes: 2, 3, 6 or 7.
  3. The main prize can be found in an envelope whose number is larger than 3.
  4. If Clue 1 is false, then Clue 3 must be true.
  5. Both Clue 2 and Clue 4 are true.

Which envelope should Cara choose, and why?

This puzzle is an example of the use of logic in solving mysteries. Logic is more than just a mystery-solving device, though. Thanks to the work of mathematician George Boole – whose birthday we celebrate every 2 November – it is also fundamental in the design and functioning of any electronic device today.

The Boole2School Legacy project, initiated by University College Cork in partnership with Maths Circles Ireland, provides playful lessons on Boolean logic, which can be downloaded by teachers and enjoyed by school students aged between 7-18.

Scroll down for the solution to this week’s puzzle.

Prize envelope

The €1m prize envelope. Image: graja/Shutterstock


“Frank, I’ll take envelope five, please.”

Cara was right, of course, but how did she get there?

First, she examined the last two clues.

If Clue 5 were true, then Clues 2 and 4 would also be true. But, as only three of the clues can be true, that would mean that Clues 1 and 3 would have to be false. This contradicts Clue 4 which states that if Clue 1 is false, Clue 3 must be true.

Here, Cara found a contradiction; so Clue 5 must be false.

Since Clue 5 is false, at least one of Clue 2 and Clue 4 must be false. Cara knew that it was not absolutely necessary for both Clue 2 and Clue 4 to be false. One falsity would be enough to falsify Clue 5.

Cara assumed first that Clue 4 was false, which would mean that Clue 1 and Clue 3 were false. But this is impossible as only two clues were false.

So Clue 4 can’t be false, and Clue 2 must be, instead.

At this point, Cara knows that Clue 5 is false, Clue 4 is true and Clue 2 is false. As three of the clues must be true, that means that Clue 1 and Clue 3 must also be true.

Cara therefore knew that:

  1. The €1m was in an odd-numbered envelope
  2. The €1m was in an envelope with a number higher than 3
  3. The €1m was not in the envelopes numbered 2, 3, 6 or 7

Logic dictated, then, that the money must be in envelope five.

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Kirsty Tobin was careers editor at Silicon Republic