Number Inequality Problem
One fine day, 7 people (let's name them A, B, C, D, E, F, G) were playing a game. They all had a number from 1 to 7, where none of them had the same number. They were so interested in finding out which number each person received, so the following conversation among them ensued:
A: My number is odd.
B: My number is greater than 3.
C: My number is prime.
D: My number is smaller than 5.
E: My number is a square number.
F: My number is even.
G: My number is ===.
Nobody could understand what G said as he had a sore throat.
Nevertheless, if we know that:
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F's number is greater than A's
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D's number is greater than C's
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B's number is greater than E's, and
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The sum of A, B and C's numbers = The sum of D, E, F and G's numbers
Find the product of D, E and G's numbers.
The answer is 12.
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1 solution
By looking at the information the question have, we have:
A = 1/3/5/7
B = 4/5/6/7
C = 2/3/5/7
D = 1/2/3/4
E = 1/4
F = 2/4/6
Since F > A, A cannot be 7.
Since D > C, D cannot be 1 or 2, C cannot be 5 or 7.
Now, we have:
A = 1/3/5
B = 4/5/6/7
C = 2/3
D = 3/4
E = 1/4
F = 2/4/6
The sum of 1 to 7 is 28, and half of that is 14.
Note that the highest possible sum for A, B and C is 5 + 7 + 3 = 15!! This is only one short from our target, 14!!
Hence, we must subtract 1 from either A, B or C.
So, A is 5, because the 2nd highest possible A must subtract 2 from the highest possible A, and B cannot be 4 or 5.
We are doing some progress! Let's see what we got:
A = 5
B = 6/7
C = 2/3
D = 3/4
E = 1/4
F = 2/4/6
Since F > A, F is 6. B cannot be 6 now, so B is 7.
As neither A or B subtracted 1 from their highest possible number, the 'subtract 1' award goes to C!!
C is therefore (3 - 1) = 2
In the end, we have:
A = 5
B = 7
C = 2
D = 3/4
E = 1/4
F = 6
G = 1/3/4
Do we have to do any more complicated calculations?! NO!!
The only remaining digits left unused are 1, 3 and 4. All we have to do now is find their product, which is:
1 x 3 x 4 = 12