Challenges, Hints and Answers 
Damsel in Distress ^{*} Once upon a time, a notorious knight captured a damsel and imprisoned her in a castle surrounded by a square moat that was infested with extraordin arily hungry alligators. The moat was 20 feet across, and no drawbridge existed because after depositing the damsel in the castle, the evil knight had taken it with him. After a good time, a good knight rode up and said, “Hail sweet damsel, for I am here, and thou art there. Now what are we going to do?” The knight, though good, was not too bright and consequently paced back and forth along the moat looking anxiously at the alligators and trying feebly to think of a plan. While doing so, he stumbled upon two sturdy wooden beams suitable for walking across, but lacking sufficient length. Alas, the moat was 20 feet across, but the beams were each only 19 feet long and 8 inches wide. He had no nails, screws, saws, Superglue, or any other method of joining the two beams to extend their length. What to do? What do to?? Challenge: Help the knight rescue the damsel in distress using only the two beams of wood that were found. Once you believe that you’ve found a solution, show it to the challenge captain. If you are correct, then you will receive your next clue.
Hint: Look at the corner of the square moat! Answer: Use one beam to span the corner and the second beam to cross onto the ground.

Fountain of Knowledge ^{*} While on safari, Trey Sheik suddenly found himself alone in the Sahara Desert. Both hours and miles passed as he wandered aimlessly through the desert, but he stumbled upon an oasis as he was nearing dehydration. There, sitting in a shaded kiosk beside a small pool of mango nectar, was an old man named Al Dente. Big Al not only ran the mango bar but was also a travel agent and could book Trey on a twohumped camel back to Kentucky. At the moment, however, Trey desired nothing but a large drink of that beautifully translucent and refreshing mangoade. Al informed Trey that he sold the juice only in 8ounce servings and the cost for one serving was $3.50. Trey frantically searched his pockets and discovered that he had exactly $3.50 (along with a lot of sand). Trey’s excitement was soon shattered when Al casually announced that he did not have an 8ounce glass – all he had was a 6ounce glass and a 10ounce glass, both with no markings on them. And, Al was quite adamant and would only sell Trey exactly 8ounces of juice. Challenge: Determine how Trey can get exactly 8ounces of juice in the 10ounce glass using only the 6ounce and 10ounce glasses. Once you believe that you’ve found a solution, show it to the pit stop Captain. If you are correct, then you will receive your next clue.
Hint: Try filling up the 10ounce glass, and then use it to fill the 6ounce glass. What do you have now? Answer: Fill the 10ounce glass and pour juice from this glass into the 6ounce glass until it is full – this leaves 4 ounces of juice in the 10ounce glass. Next, empty the 6ounce glass and transfer the 4 ounces of juice in the 10ounce glass into the 6ounce glass Fill the 10ounce glass with mango juice and pour juice into the 6ounce glass until it is full. Since it already had 4 ounces of juice in it, you can only add 2 ounces. This leaves you with 8 ounces of juice in the 10ounce glass. An alternate solution is to first fill the 6ounce glass with juice and pour it into the 10ounce glass. Then fill the 6ounce glass again and pour 4ounces into the 10 ounce glass. Dump the juice out of the 10ounce glass. Transfer the 2 ounces of juice remaining in the 6ounce glass to the 10ounce glass. Then, fill the 6ounce glass and pour its entire contents into the 10ounce cup. This results in 8 ounces of juice in the 10ounce cup.

Cornhole … Can you spot a sequence? In the game of corn hole, points are scored according to the following rules: 5 points for in the hole 3 point for on the board 0 points for off the board Challenge: Find the formula for the nth term of the sequence and show the formula to the Pit Stop Captain. If you are correct, the Captain will give you a value for n. You are to determine the number of rectangles in the nth term of the sequence for the given value of n and score at least that many points in a game of Bean Bag Toss.
Answer: The formula for the nth term of the sequence is f(n) = 3n  2. The Pit Stop Captain should ask each team to find the 12th term of the sequence. The twelfth term of this sequence is f(12) = 3(12)  2 = 34.

Tower of Hanoi The objective of the puzzle is to move the entire stack of disks to another rod, obeying the following rules

Only one disk may be moved at a time.

Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.

No disk may be placed on top of a smaller disk.
Challenge: Solve the Tower of Hanoi problem with three disks. Once you have a solution, show your challenge captain. If you are correct, then you will receive your next clue.
Hint: Move the smallest disk to the third post and the middle disk to the second post. Answer: 7 moves
Move 1: move disk 3 to post C
Move 2: move disk 2 to post B
Move 3: move disk 3 to post B
Move 4: move disk 1 to post C
Move 5: move disk 3 to post A
Move 6: move disk 2 to post C
Move 7: move disk 3 to post C

Meanie Genie ^{*} On an archeological dig near the highlands of Tibet, Alley discovered an ancient oil lamp. Just for laughs she rubbed the lamp and was shocked when an ornery genie named Murray appeared. Begrudgingly, the genie granted Alley three wishes. Thinking quickly, she said “I’d like to find the Rama Nujan, the jewel that was first discovered by Hardy the High Lama.” Murray agreed, and instantly nine identical looking stones appeared. Alley looked at the stones but was unable to distinguish one from the others. Murray finally told her that the Rama Nujan was embedded in one of the stones, but she could only take one with her, so she needed to make her choice wisely. Murray then let her in on a hint – eight of the stones weigh the same, but the stone containing the jewel weighs slightly more than the others. In exasperation, Alley wished for a scale under her breath, and ... poof ... a scale appeared. Unfortunately, there was a trick to the scale – it could only be used once. Now she was down to a single wish. As she began thinking out loud, she accidentally wished for another scale, which magically appeared. Again, though, this scale could be used only once. Challenge: If Alley may use each of the balance scales only once, explain how she can select the stone containing the Rama Nujan stone. Once you have a solution, show your challenge captain. If you are correct, then you will receive your next clue.
Hint: Instead of comparing stones individually, compare one group of stones to another group of stones. Answer: Arrange the set of 9 stones into three groups of 3 stones each. On the first scale, weigh two groups against each other. If they have the same weight, then the heavier stone is in the third group that was not weighed. If one set is heavier, then the heavier stone is in this group. Now that we know which group contained the heavier stone, we can compare two stones against each other on the second scale. If they have the same weight, then the stone containing Rama Nujan is the unweighed stone. If one of these two stones is heavier, then it must contain Rama Nujan.

A Fox, a Goose, and a Bag of Beans Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases  the fox, the goose, or the bag of the beans. If left alone, the fox would eat the goose, and the goose would eat the beans. The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it? Challenge: Determine how the farmer was able to get all his purchases to the other side of the river. Once you have a solution, show your challenge captain. If you are correct, then you will receive your next clue.
Hint: Give the first step in the solution. Answer: His actions in the solution are summarized in the following steps:

Bring goose over

Return

Bring fox or beans over

Bring goose back

Bring beans or fox over

Return

Bring goose over
Thus there are seven crossings, four forward and three back.

GIANT Tangrams Challenge: Put the tangram puzzle pieces together to make one of the following pictures:
Hint: Show them the location of 23 Tangram pieces. Answer: The pictures below show the solutions for the two shapes given above.

21 Flags There are 21 flags placed in the ground. You must strategize to select one, two, or three flags to remove at a time, alternating turns with the other team. The racing team will choose whether they want to take the first turn or not. Challenge: Win the game by being the team that removes the final flag(s). To receive a checkmark on your passport, you must win 2 games of 21 Flags out of a maximum of three games. If you lose two games, then you’ll receive an X on your passport for this challenge.
There is no hint or solution for this particular challenge  team members must work together to formulate a strategy that will result in their winning the game.

Cornhole … Working with dimensions Challenge: Find the surface area of the top of the Cornhole board. Your challenge captain will provide a tape measure and calculator for your use. The Challenge Captain will verify that your solution is correct. Once you have the correct solution (within 0.1 ft^{2}), you must score at least that many points in a game of Cornhole where each bean bag that goes in the hole is worth 2 points and each bag on the board is worth 1 point.
Answer: If a standard cornhole board is used, the surface of the top of the board is 7.8 ft^{2} – any area between 7.7 and 7.9 ft^{2} will be acceptable. A minimum of 8 points must be scored in the game.

Army Men Your Challenge Captain has a box containing an unknown number of army men. Each plastic figure in the box has been numbered consecutively on its base. Challenge: Determine how many men are in the box by taking samples according to the following guidelines: First, take a sample of size 2 army men – come up with a conjecture as to how many men are in the box based on this sample and tell your Challenge Captain. If your answer is correct (within 10 men), then you will receive a checkmark on your passport. If your answer is incorrect, place the sampled army men back into the box and remove a sample of 4 army men. Again, come up with a conjecture as to how many men are in the box and tell your Challenge Captain. If your answer is correct (within 10 men), then you will receive a checkmark on your passport. If your answer is incorrect, place the sampled army men back into the box and remove a sample of 6 army men. You may continue this sampling technique up to a maximum of 10 army men. If you are incorrect on your fifth conjecture, then you will receive an X on your passport.
The solution will depend upon how many army men are placed in the box. For our race, a total of 215 men were consecutively numbered.

Mathdoku Challenge: Complete the Mathdoku puzzle given below by filling in each row and column with the digits 1 through 4 (no repeats allowed) in such a way that in each bold outlined region, the given mathematical operation yields the given number.
Hint: Give the participants any two of the squares. Answer: Solution given below.

Perfect Egg You find yourself in the kitchen of a restaurant where you have been charged with the task of boiling an egg for exactly fifteen minutes. Unfortunately, there is no kitchen timer – there is only a 7minute hourglass and an 11minute hourglass. Challenge: Using only these two hourglasses, what is the quickest way to time the boiling of an egg for exactly fifteen minutes?
Hint: Give the participants the first two steps in the best answer given below. Best Answer:

Turn both timers over at the same time.

Let the 7minute timer run out, then immediately flip it back over.

The 7minute timer has reset, but four minutes of sand remaining in the 11 minute timer.

7 minutes have elapsed.

When the 11minute timer runs out, immediately flip the 7minute timer back over.

11 minutes have elapsed.

Four minutes of sand had passed through the 7minute timer; flipping it back over will give an additional four minutes.

15 minutes have elapsed.
Not the Best Answer – no checkmark awarded (may try again)

Turn over both timers at the same time.

Let the seven minute time run out.

Start boiling the egg now.

Then, when that four minutes elapses, turn the eleven minute timer over again (4 + 11 = 15) and let the egg boil.
Note: this solution results in a total time of 22 minutes (7 + 4 + 11), which is not the quickest way to time the egg boiling. Emphasize that there is a faster way!

Note: Challenge titles denoted by an asterisk (*) were taken from: Burger, Edward B, and Michael Starbird. 2005. The heart of mathematics: An invitation to effective thinking. Emeryville, CA: Key College Publishing. 
