**Puzzle 11**

If the areas of three sides of a rectangular box are 24, 32 and 48 square inches, respectively, what is the volume of the box?

Show Solution*A*,

*B*and

*C*represent the dimensions of the box. Given: 1.

*AB*= 24 2.

*BC*= 32 3.

*AC*= 48. Divide Equation 2 by Equation 3 to get

*B/A*= 32

*/*48 = 2

*/*3, so

*B*= 2

*A/*3. Substitute into Equation 1 to get

*A*(2

*A/*3) = 24, so

*A*² = 36,

*A*= 6,

*B*= 4,

*C*= 8. Volume = 6•4•8 = 192

**Puzzle 12**

I was at a party and the energetic 92-year-old host told me that he had three children. When I asked how old they were, he said, “I’ll give you a hint; the product of their ages is 72.” I said, “There are too many ways to factor 72. I need more information.” He said, “The sum of their ages is the same as my house number.” I went to the front door, opened it and looked at the number. “Still not enough info,” I said. “OK,” he said, “my oldest child loves chocolate pudding.” “Now it’s obvious,” I said. “Your children’s ages are ___, ___ and ___.” Fill in the blanks. And no, you don’t need to know the house number to get the answer, but you do have to explain how each piece of information is used to get the answer.

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**Puzzle 13**

In Square *PQRS*, *M* is the midpoint of side *PQ* and a segment joining *S* to *M* is drawn. A perpendicular is drawn from *R* to segment *SM* and intersects it at *T*. If a side of the square measures 2 feet, what is the area of Quadrilateral *MQRT* ?

**Puzzle 14**

AAAA + BBBB + CCCC = BAAAC Each distinct letter represents a different digit and every occurrence of the same letter represents the same digit. What must each letter represent for this to be a correct addition? (No guesswork allowed. There are 720 ways of assigning values to these three letters and reasoning about arithmetic will eliminate all but one of them.)

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**Puzzle 15**

Prove that the product of four consecutive positive integers cannot be a perfect square. (This is not so much a puzzle as a challenge to high school algebra students.)

Show Solution*n*

*+*1,

*n*+ 2,

*n*+ 3 and

*n*+ 4, with

*n*≥ 0, represent the four consecutive positive integers. Their product is (

*n*+ 1)(

*n*+ 2)(

*n*+ 3)(

*n*+ 4). Multiplying the second and third factors together and the first and fourth factors together yields (

*n*

*² +*5

*n*+ 6)(

*n*

*² + 5n*+ 4) = [(

*n*

*² +*5

*n*+ 5) + 1] times [(

*n*

*² +*5

*n*+ 5) – 1] = (

*n*

*² +*5

*n*+ 5)² – 1. Since two consecutive positive integers cannot both be perfect squares and (

*n*

*² +*5

*n*+ 5)² is a perfect square, (

*n*

*² +*5

*n*+ 5)² – 1 cannot be a perfect square.