#### Puzzle 1

Replace the letters in this pattern with the numbers 2 through 6 so that no number is used more than once and B + C + D = A + C + E.

What did you do to solve this puzzle?

Why?

How many distinct solutions are there? What property must the number that replaces the letter C have?

Why?

Show Solution

#### Puzzle 2

If each dot in the diagram on the left represents a coin, then we have arranged 20 coins into five rows (line segments) of four coins each.

In the diagram on the right, we have arranged 15 coins into five rows of four coins each.

Is it possible to arrange only ten coins into five rows of four coins each?

As with most possibility questions, a “yes” answer requires showing *how* to do it, and a “no” answer requires a *proof* that it can’t be done.

#### Puzzle 3

A very large swimming pool has eight drains. If all but the first drain are operating, the pool drains at the rate of 169 gallons/minute.

If all but the second drain are operating, the pool drains at the rate of 178 gallons/minute.

All but the third: 171 gal./min. All but the fourth: 172 gal./min. Fifth: 177.

Sixth: 184. Seventh: 173. Eighth: 176.

At what rate will the pool drain if all eight drains are operating simultaneously?

Show SolutionAt this point, we will abandon the standard method, which would involve either using matrices or eliminating all but one of the variables the way we do with systems of two or three equations. Either one of these methods would be complicated, messy and tedious. Rather, we notice the each variable is missing from exactly one equation, so we add the eight equations together, yielding the equation, 7A+7B+7C+7D+7E+7F+7G+7H=1400. Dividing both sides by 7, we get A+B+C+D+E+F+G+H=200. So the rate we are seeking is 200 gallons/minute. A second solution would be to let *x* equal the rate of the first drain and then express all the other rates in terms of the first rate. Since the pool drains most slowly when the first drain is not operating, the rate of each of the other drains is less than that of the first. The second through eighth rates would therefore equal *x*-9, *x*-2, *x*-3, *x*-8, *x*-15, *x*-4, and *x-7. *Adding together the second through eighth rates yields 7*x*-48, which we are told is equal to 169. Solving that equation yields *x*=31, so the sum of all the rates is 8(31)-48=200 gallons/minute. A third method would involve letting *x* equal the sum of all the rates, but I’ll leave that one to you.