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x implies y

Part I

i have a black triangle, a white triangle, a black circle and a white circle.  if i gave you a shape (triangle or circle) and a color (black or white), the "frobby" items would be those that had either the shape or the color, but not both.  that is, in order to be frobby, the item must be of the specified color OR the specified shape, but not both the specified shape AND the specified color.  i'm thinking of a shape and a color in my head and i tell you that the white triangle is frobby.  can you tell me the "frobbiness" of the other items?

Answer: Yes - the black circle is frobby and both the white circle and the black triangle are non-frobby. The thing here is that you don't need to know what the shape that you deduce frobbiness from is. It could be the white circle or it could be the black triangle but the chosen shape is non-frobby because it has both qualities and the other is non-frobby because it has neither.

Part II

there are four cards which have a letter on one side and a number on the other side.  i lay them out and the cards appear as 2 5 F E.  the rule is that a card with an odd number on one side must have a vowel on the other.  what is the minimum number of cards you should turn over to prove the rule is true (and which cards would they be)?


The answer is two and the cards are 5 (to see that the other card is a vowel) and F (to see that the other card is an odd number). The "trick" here is that you don't need to turn over E because the rule is that a card with an odd number must have a vowel but not that a card with a vowel must have an odd number.

The second ties up with the first because in both cases you are tempted to assume you need something you don't. This no doubt explains the aha! quality. And this making of false assumptions is a common trait of many brain teasers. For example the person dressed walking along the street with no street lights, moon or stars visible who is not run over by the car with no lights because it is day time, or the two trains that both enter the tunnel 200 km single track tunnel in the morning at 100 km/h each but don't crash because the problem doesn't specify the time they both entered the tunnel.

Thus practise in these kind of brain teasers is not just a cheap party piece, since it may just persuade you to challenge obvious assumptions, and more practically for a programmer to check that you have specified the obvious, since none of the brain teasers will present any problem to a computer program - it simply won't see the difficulty.

Stephen Jones
Thursday, October 24, 2002

In Part II of x implies y, the rule is that if a card has an odd number on one side, the other side of the card would have a vowel.  If you were trying to prove this rule is true, wouldn't you want to turn over the F to make sure the other side of the card was even?

CM
Thursday, October 24, 2002

Yes, Sorry a typo! You need to ensure that there is an even number on the other side of the consonant.

Stephen Jones
Friday, October 25, 2002

What if there's an odd number on the other side of
the "2" card ? Shouldn't you turn "2" over as well to
make sure this isnt the case ?

Mike
Sunday, November 10, 2002

"What if there's an odd number on the other side of
the "2" card ? Shouldn't you turn "2" over as well to
make sure this isnt the case ? "

There can't be an odd number on the other side of the "2" card because the question states that there must be a letter on one side and a number on the other.

And whether the letter is a vowel or consonant is irrelevant. The rule is "that a card with an odd number on one side must have a vowel on the other" 2 is an even number so what it has on the other side doesn't affect the rule.

Stephen Jones
Thursday, November 14, 2002

surely you only need to turn over one card, the 5, to see if it has a vowel on the other side - "x implies y" doesn't imply "not x implies not y".
If the cards are (2A, 5A, F4, E4) this is still consistent with the rule.

Matt Mitchell
Thursday, November 14, 2002

You need to turn over the F card to ensure that there is an even number on the opposite side.

E4 is compatible with the rule, as you said, but F7 would not be. The odd number 7, would require a vowel.

Stephen Jones
Friday, November 15, 2002

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