**Puzzle 76**

I took a logic class in college that met five days a week, Monday through Friday, and one Monday, at the very beginning of class, the teacher announced, “In one class meeting this week, I will give you a surprise quiz.” Since this was a logic class, we were all able to figure out which was the only day he could give us the quiz. Which day was it? (It’s not enough to show why your answer works; you must also show why each of the other four possible days does not work.)

Show Solution*surprise*quiz on Wednesday or on Tuesday. This leaves Monday as the only day of the week that he could give us a

*surprise*quiz, that is, a quiz that we didn’t know about before the class meeting in which the quiz was to be given.

**Puzzle 77**

The numbers, 3, 4, 5, 6 and 7, are typed and printed onto five distinct cards, arranged in an addition column, and the same is true for the numbers, 8, 7, 6, 3 and 2. Exchange one card from the first column with one card from the second column in such a way as to get the same sum in both columns.

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**Puzzle 78**

This one is for high school geometry students. Without using Heron’s Formula, which is usually not taught in a basic geometry course, anyway, find the area of a triangle whose sides measure 13, 14 and 15.

Show Solution*b*is the measure of the base and

*h*is the height, i.e., the measure of the altitude drawn to that base. Draw the altitude of the triangle to the side measuring 14, thereby creating two right triangles with a common altitude. Call the length of this altitude

*h*and the measures of the bases of the two right triangles

*x*and 14 –

*x.*Applying the Pythagorean Theorem to each of the two right triangles, we get and . Solving this system of simultaneous equations, we get

*x*= 5 and 14 –

*x*= 9, and one more application of the Pythagorean Theorem yields

*h*= 12, so the area of the original triangle is 84.

**Puzzle 79**

You are in a race and you overtake the person who is in first place. What place are you in, now? OK, so you got that one right away. How about this one: You are in a race and you overtake the person who is in last place. What place are you in, now?

Show Solution & Corollary Puzzle

**Puzzle 80**

Find unequal rational numbers, * a* and * b* (other than 2 and 4), such that .

*a*and

*b*are both rational, let

*b*=

*ra*, where

*r*is rational. Then , , , . Since

*r*is rational,

*a*is rational whenever is an integer, i.e., whenever

*r*– 1 is the reciprocal of an integer, so , where

*k*is an integer, so . Therefore, and . These values for

*a*and

*b*satisfy the given equation for

*k*equal to any non-zero integer, so there are an infinitude of solutions. For example, for

*k*= 2, and , so . And in general, if and , where

*k*is a non-zero integer, then

*a*and

*b*are both rational, and