This was a Challenge Problem last summer at our Bard College site. Challenge Problems encourage students to work collaboratively; once it's solved, the whole program gets a prize. (For this one, a sleepover on the last night!) So now it's your turn... can you solve it?
Malachi takes 42 pieces of candy, each of which is either a Skittles or an M&M, and arranges them in a circle letter side down (so they look identical!). He tells Hodaya that 23 of the candies are next to at least one Skittles, and 35 are next to at least one M&M. If Hodaya can figure out how many Skittles and how many M&Ms there are, she gets all of them!
How can Hodaya figure this out? (The answer appears below the image.)
Solution:
The important thing is to go step-by-step to figure out what you can.
If there are 23 candies next to at least one Skittles, that means 42-23=19 candies are not next to any Skittles — so they must be next to two M&Ms.
If 19 candies are next to two M&Ms and 35 are next to at least one M&M, that means 35-19=16 are next to exactly one M&M.
Now here's the key insight. Suppose there are x M&M's. Each M&M has a neighbor on the left and on the right, so if you add up for all the pieces how many M&M neighbors they have, you should get 2x. Well, we know that 19 candies are next to two M&Ms, so they have a total of 38 M&M neighbors. And 16 are next to one M&M, so they have 16 M&M neighbors. That gives 54 total M&M neighbors.
Divide by two to get 54/2=27 M&Ms in total.
With 42 candies in total, subtract out the M&Ms to get 42-17=15 Skittles.
Sometimes, it's amazing how much you can figure out with what seems like too little information!