AMC 10 · 2013 · #10
Grade 6 logicnumber-theoryPick an answer.
Following every trade one by one would take dozens of steps. Instead, re-describe the tokens by giving each color a point value chosen so that every trade keeps the total number of points the same. Then the total is fixed from start to finish, and the only thing left to pin down is how the leftover points split among the few tokens that remain at the end.
Find each trade's net effect
Each trade has a simple net effect.
Boiling each booth down to a plus-and-minus list makes the two trades easy to compare.
4.OA.A.3Organize Information In More WaysGive each token a point value
Point values make both trades value-neutral.
If both booths trade equal points for equal points, the total point value can never change.
If both booths trade equal points for equal points, the total point value can never change.
▸ Why?
What is handed in is exactly matched by what is handed back, so nothing is created or destroyed.
▸ Why?
The total is the tokens' point values added together, so a fixed total ties the start to the end.
Compute the fixed total
The fixed total comes to 525.
One fixed number, 525 points, ties the messy start to the messy end.
4.NBT.B.5Organize Information In More WaysDescribe the end state
The end state leaves very few leftovers.
Only a tiny handful of red and blue tokens can survive to the end.
6.NS.C.7Work BackwardsSplit the 525 points at the end
Splitting the total leaves only two options.
The silver count must be whole, so only leftovers that are multiples of 5 can actually happen.
4.OA.B.4Extreme PrincipleRule out zero-zero and finish
One is impossible, so the answer is 103.
The last trade always returns a coin, so you can never empty both colors at once.
4.OA.A.3Extreme PrincipleGive each token a point value so every trade is an even swap, then the unchanging total tells you the answer no matter how the trades happen.
- Find each trade's net effect
- Give each token a point value
- Compute the fixed total
- Describe the end state
- Split the 525 points at the end
- Rule out zero-zero and finish