### Brain Teasers

# Parakeet Basketball

Your favorite college basketball team, the Fighting Parakeets, is playing against a rival, but you haven't been able to see any of the game yet because you have been busy. However, your brother was in a nearby room watching it. Your brother sees a statistics display during the halftime break and decides to give you a trick math test. You come into the room as the second half is beginning and see that your team is winning 32-29. Your brother says, "The Fighting Parakeets have scored on 18 shots. They have made at least twice as many 2-point shots as free throws. How many free throws, 2-pointers, and 3-pointers have they made?" He thinks you will become confused in your figuring because he has not given you quite enough information to find the solution. However, you could hear bits and pieces of the game from the next room while you were busy, and you happened to hear that your favorite Fighting Parakeets player nailed a long 3-point shot. What is the answer to your brother's question?

### Hint

A free throw scores 1 point.### Answer

5 free throws, 12 2-point shots, and 1 3-point shot.Let x be the number of free throws made,

y be the number of 2-pointers made,

and z be the number of 3-pointers made.

From the score, you know (A): x + 2y + 3z = 32

From the quote, you know (B): x + y + z = 18

Multiply (B) by 3 to get (C): 3x + 3y + 3x = 54

Subtract (A) from (C) to get: 2x + y = 22, or (D): y = 22 - 2x

From the quote, you also know (E): 2x <= y

Substituting (D) into (E) gives: 2x <= 22 - 2x, or (F): x <= 5.5

If x=5, then from (D) you get y=12, and from (A) you get z=1.

If x=4, then y=14, and z=0.

If x=3, then y=16, and z=-1. This is not possible because z cannot be negative. If you continue decrementing x, then z will continue to decrease too.

So, you have two options that fit the criteria given by your brother:

5 free throws, 12 2-point shots, 1 3-point shot, or

4 free throws, 14 2-point shots, 0 3-point shots.

So it's a good thing that you heard that your favorite player made a 3-pointer, because that eliminates the second option!

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