Thinking: Solving a Substitution Puzzle

Recently BrainBashers offered this puzzle: If Poland is 44 miles away, Canada is 42 miles away and Mongolia is 56 miles away, how far away is Peru?

Try it yourself before reading on, I’ll wait.

No, I’m serious, try it.

It’s not that difficult.

Basic algebra.

Eighth grade? Ninth grade? When is algebra taught anyway?

Oh good, you’re back. Here’s how I solved it.

Assume the game is assigning a value to each consonant and vowel.

Poland = 4 consonants and 2 vowels so: 4c + 2v = 44
Canada = 3c + 3v = 42
Mongolia = 4c + 4v = 56

You want to solve for one variable, so look to see what you can substitute equally. Mongolia and Poland both have a 4c, so solve for that.

4c + 4v = 56
4c + 4v – 4v = 56 – 4v
4c = 56 – 4v

Substitute the 4c value into Poland, and then solve for c, which gives you:

4c + 2v = 44
(56 – 4v) + 2v = 44
56 – (4v + 2v) = 44
56 – 2v = 44
56 – 2v + 2v = 44 + 2v
56 = 44 + 2v
56 – 44 = 44 – 44 + 2v
12 = 2v
12 / 2 = 2v / 2
6 = v

Then substitute 6 for v into any formula to get the value of c.

4c + 2v = 44
4c + 2(6) = 44
4c + 12 = 44
4c + 12 – 12 = 44 – 12
4c = 32
4c / 4 = 32 / 4
c = 8

With c = 8 and v = 6 the value of Peru is:

Peru = 2c + 2v
2(8) + 2(6)
16 + 12
28