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Mon 14 Apr, 2008 08:13 am
George and Harry enjoy creating and solving puzzles. They also enjoy playing card games. On one occasion, George was shuffling the cards, a standard deck, when three cards fell out and landed face-down on the table. George picked up the cards and was about to return them to the deck when he noticed an interesting fact about them.
He said, "I have three cards here, Harry. As luck would have it, if I take the numerical value of each card and multiply the three values together, the result equals my lucky number. The highest card is a seven. What are the other two cards?"
Harry considered the problem for a few seconds and replied, "I suppose that an ace has a value of one, so each of your other two cards has a value lower than seven and possibly as low as one, right?"
"Right," confirmed George.
"Well," added Harry, "you haven't told me what your lucky number is, so I could only guess what the two cards are. I need more information to work it out."
"I see your difficulty," replied George, "so I will give you one more clue: the lowest card is a spade."
Harry pondered the problem again for a while, and then correctly identified the two cards. What were they?
How about a 4 and a 1 (ace)?
Great , you got it. How did you do that? Did you know the answer?
If the lowest card is a spade, there has to be a low card---that is the remaining two cards can't be the sam, eventhough the product must be a square number--that is 6*6=36, 5*5=25, 4*4=16, 3*3=9, 2*2=1, & 1*1=1.
With the product of the remaining two cards equal to a square the squares need to be factored.
36 is the same as , 1*36, 2*18, 3*12, 4*9, & 6*6 but in all cases one of the remaining 2 is greater than 7,
25 is the product of 1*25, and 5*5 (no other combinations is possible),
16 is the product of 1*16, 2*8, & 4*4 (16 & 8>7),
9 is 1*9, or the product of 3*3, (9>7)
4 is 1*4 or 2*2 (now we're somewhere since neither 1 or 4 is greater than 7),
1 is 1*1.
The only square number that fits is 4 with products 1 and 4.
And 1 is the ace of spades.
BTW his lucky number is a 'perfect' 28.
Rap
that is ok but the lowest card is of spade so can't we take 1 and 3 or 4 and 5?
The clue is that there is a lowest card. That it is a spade is extra not really pertinent.
Rap
I did it as rap showed. You could make an argument for a variety of answers based on the clues given. The assumption is that since several combinations match the clues, the fact that the two cards aren't the same (there is a smallest card) must enable one to to decide between two of the combinations with the same product.
Actually I was working in several directions, but when you said markr was right, it became obvious to me what direction to take---that the product of the two unknown cards was a perfect square.
A perfect number being the lucky number (7*4*1) adds icing to the cake.
28 factors into 1, 2, 4, 7 & 14 & 28=1+2+4+7+14--- viola a perfect number. My favorite numbers are also Perfect.
Rap
and why can't his lucky number be 35???
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