Question

Directions: Using the digits 1 to 9, at most one time each, fill in the boxes to find the largest or smallest possible values for the sum of X and Y.

Answer 1

Smallest: 8x-3=4; 1y+9=2. total= -49/8

Largest: 1x-9=8; 2y+3= 7. total= 19

Explanation:

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Answer 2

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Answer:

smallest: 8x -3 = 4; 1y +9 = 2. total = -49/8

largest: 1x -9 = 8; 2y +3 = 7. total = 19

Explanation:

If we use variables to represent the box contents, we can write ...

• ax -b = c
• dy +e = f

Then the values of x and y are ...

x = (c +b)/a

y = (f -e)/d

For positive integer values of the variables, x will always be positive, and y may or may not be negative.

Smallest sum

For the sum to be the smallest, we must have x be as small as possible and the ratio (f-e)/d be as negative as possible.

x will be small for large 'a' and for (c+b) small. For y to be as negative as possible, we want 'd' and 'f' small and 'e' large. Best results are obtained for

• 8x -3 = 4 ⇒ x = 7/8
• 1y +9 = 2 ⇒ y = -7

For these coefficients, the sum is -6 1/8 = -49/8.

(note that the values of 'b' and 'c' can be swapped with no net effect)

Largest sum

For the sum to be the largest, we must have x as large as possible: (b+c) large and 'a' small. At the same time we must have y be positive and as large as possible: (f-e) positive and large, 'd' small. Best results are obtained for

• 1x -9 = 8 ⇒ x = 17
• 2y +3 = 7 ⇒ y = 2

For these coefficients, the sum is 19. Again, 'b' and 'c' can be swapped with no effect.

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Additional comment

These extreme values are verified by examination of the 60,480 possible permutations of the coefficients.