HINTS AND SOLUTIONS
BUILDING BUILDINGS
Octahedron Net
MANCALA
We double the extra number of seeds because we need the extra to fill up the missing amount on the other side. We need twice this number, however, to know how much the player won by because before they can fill in the missing number for the other player, they have to fill in their own.
So why is the number a player won by always even? Well, the total number of seeds in the game is an even number, so when you add up the seeds each player has in their bank at the end, it must be an even number. Two numbers add to an even number if they are both odd or both even. Similarly, an even number minus an even number is an even number, and an odd number minus an odd number is an even number. Using this logic, either both players had an odd number of seeds at the end of the game and thus the difference is even. Or they both had an even number and the difference is still even. Therefore, the number of seeds a player wins by must be even!
BINARY BRACELET
We don’t need to use delimiters because the code for each letter has exactly seven 0s or 1s. So we know a new letter starts after seven 0s and 1s.
Look at the codes for a few matching uppercase and lowercase letters. What is the same? What is different?
- Every letter starts with a 1.
- Uppercase letters have a 0 next.
- Lowercase letters have a 1 next.
- The last four digits of each uppercase and lowercase letter match.
The secret message is: Math is fun.
THE PRINCESS’S PUZZLES
The princess asks one brother, “Is that one older than the other?” This question guarantees she will get either the oldest or the youngest. Why?
If she asks the oldest brother whether the first one she points to is older than the other, he will say yes if she pointed at the middle one first, and no if she pointed at the youngest first. If he says no, she should choose the one she pointed to first. If he says yes, she should choose the one she pointed to second.
If she asks the youngest brother whether the first one she pointed to is younger than the second, he will say no if she pointed to the older one, and yes if she pointed to the middle one first. If he says yes, she should pick the one she pointed to second, and if he says no she should pick the one she pointed to first.
Either way, if the prince says yes, she should pick the one she pointed to second and if he says no, she should pick the one she pointed to first. This guarantees she doesn’t end up with the middle brother.
Why does this work? Why do we pick the one we pointed to first when we hear the answer no and the other when given the answer yes? See if you can figure it out!
NOTE: If she asked the question of the middle brother, she won’t know which is which, but because she never picks the prince she questions, she’ll end up with the oldest or the youngest anyway!
In the end, the princess realizes the best lab partner was sitting in front of her the whole time. She and the other princess partner up and both of them are truthful all the time.