Alfie is 5 years old. His mother is 28. How old will Alfie be when he is exactly half as old as his mother?

## Puzzles, Riddles, and Systematic Thinking

## Wealth Redistribution

The kingdom of Syldavia has five economic classes. Feeling one day that there is too great a disparity among the wealth of the classes, the King asks his Finance Minister to devise a plan to redistribute the wealth of the classes. The FM proposes that the wealth of the top two classes be averaged, then the wealth of classes 2 and 3 be averaged, then the wealth of classes 3 and 4 be averaged, and finally that the wealth of classes 4 and 5 be averaged—a four-step process. The King likes the sound of the plan, but suggests that the progression should be from the lowest classes upwards. (In either case, the result will be a four-class system.) Which of the two plans should the lowest class prefer? Why?

## Mike’s Age

Mike’s age right now is exactly 3 times his age 3 years hence minus 3 times his age 3 years ago. How old is Mike right now?

## Jetta vs. Corvette

A Jetta and a Corvette travel the same distance from Columbus to Cincinnati. The Jetta travels half of the **distance** at *u* miles per hour and the other half at *v* miles per hour. The Corvette travels half of the **time** at *u* miles per hour and the other half at *v* miles per hour. Which car gets to Cincinnati first?

## Racing Cars on Figure-8 Track

Five racing cars are tearing along on a figure-8 track that has a total length of 7 miles. The cars A, B, C, D, and E have speeds, respectively, of 60, 72, 75, 80, and 96 mph. High walls are located along each side of the track; and so, should any two cars meet at the center, they will collide. Each driver is wondering, “How did I get myself into this? Who *is* Stella?” If the cars started at the center, all going the same way, will two cars eventually collide? If so, which two, and after how many miles and how many minutes?

## The Rocket Sled

Your science class is on a field trip to a rocket testing site. A rocket sled traveling 900 meters per second on a track passes the place where you are standing. Suddenly, 270 meters farther down the track the rocket explodes quite unexpectedly. If you hear the sound of the explosion 1.1 seconds after the sled passes by, you can calculate the speed of sound. So do it.

## Maggie & Aggie Driving

Maggie and Aggie synchronize their watches, rehearse their plan one last time, and start driving their cars at exactly midnight. Maggie heads due east, and Aggie heads due north, traveling 15 mph faster than Maggie. At exactly 1:20 a.m. the two ladies are exactly 100 miles apart, as observed from an inconspicuous police glider piloted by Sulphronia the super agent, who is watching through infra-red binoculars. At what speeds are the two sisters driving?

## NewYork – Washington Trains

Suppose that Amtrak has trains leaving from Washington to New York City and also from New York City to Washington every hour on the hour. Also suppose that the trip from one city to the other takes 4.5 hours and all trains travel at the same speed. If you are on the train from DC to NYC, how many trains going the other way will you pass?

## Factor Fourth Degree Equation

The number of distinct ordered pairs (*x*, *y*), where *x* and *y* have positive integral values satisfying the equation *x*^{4}*y*^{4} – 10*x*^{2}*y*^{2} + 9 = 0, is:

- 0
- 3
- 4
- 12
- infinite

## Progression to Quadratic

Consider the following interesting list of integers:

*P*

_{0}= 41,

*P*

_{1}= 43 = 41 + 2,

*P*

_{2}= 47 = 43 + 4,

*P*

_{3}= 53 = 47 + 6, … ,

in which *P*_{n} is obtained by adding 2*n* to *P*_{n-1}.

It so happens that there is a quadratic function *F*(*x*) with the property that *P*_{n} = *F*(*n*)*n*. Find a formula for *F*(*x*).

Once you have the formula, consider the question of whether or not *F*(*n*) is a prime number for every nonnegative value of *n*.