19. Cards and Conditional Verification
Five cards lie on a table. You can see only the upper face of each card and the numbers showing are:
1 2 3 4 5
Each card has a (possibly different) positive integer printed on its reverse side.
We want to test the statement:
“If a card has a 2 on one side, then it has a 5 on the opposite side.”
What is the smallest number of these five cards you must turn over, and which ones, in order to determine with certainty whether the statement is true for all five cards?
Added 20 April 2009
Hint: Ask yourself: which visible cards could possibly violate the rule? Those are the only cards you need to check.
Solution:
The rule can be violated in two ways:
- A card shows 2 on the face we see, but the hidden side is not 5.
- A card shows something other than 5 on the face we see, but the hidden side is 2.
Therefore we must turn over:
- The card showing 2, to confirm that its reverse side is 5.
- Every card that does not show 5 — that is, the cards showing 1, 3 and 4 — to be sure none of them hide a 2.
We do not need to turn over the card showing 5, because regardless of what is on its back it cannot contradict the rule.
Thus the minimum is four cards: 2, 1, 3, and 4.
Comments (3)
4, you need to know that the 1,3, and 4 all don't have a 2 on the other side. you also need to know that the 2 has a five on the other side. You don't need to flip the 5 because the converse of implication need not be true
2 because it doesn't matter what's on the other side of 1,3, and 4. You need to know if there's a 5 on the other side of the 2 and a 2 on the other side of the five. Once you've figured that out you don't need to keep flipping because no matter what's on the other side of the 1,3, and 4 there most certainly isn't a 2 or a 5 on the side you can see.
bds021 is correct.
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