Puzzles about Money and Water Balloons
OK, are you hooked? No? Well what if I told you these problems are secretly about balanced ternary, a base system with negative digits?
I realize that sounds exciting for about 0.1% of people on Earth, if I'm being generous. But if that's you, give them a try!
If you want a printable version of this problem sequence, click here. For the solutions, click here. $\def\M{\operatorname{\mathbb{M}}} \def\qty#1#2{#1~#2} \def\oz{\operatorname{oz}}$
Problem 1
You’re on vacation in Mathville. Your math teacher wanted you to bring back a souvenir, so they gave you some Mathville $\M$arks to spend. You have:
- Some $27\M$ bills
- Some $9\M$ bills
- Some $3\M$ bills
- Some $1\M$ bills
You find some chalk that you think your teacher would love. The chalk costs $19\M$. What combination of bills should you use to buy it?
Problem 2
Now that you’ve bought that chalk, you realize you want a souvenir for yourself. After your last purchase, you only have one bill of each size left in your wallet. So you have a single $27\M$, a $9\M$, a $3\M$, and a $1\M$ bill left. The cashier can give you change, but she also only has one of each bill. With just these bills, is it still possible to buy yourself exactly $19\M$ of chalk?
Problem 3
You change your mind and want a more expensive souvenir for yourself. You still have one of each bill and the cashier also has just one of each. How much is the cheapest souvenir that you would not be able to buy yourself exactly?
Problem 4
A generous stranger walks in and gives you an $81\M$ bill. She also gives one to the cashier. Does your answer to problem 3 change? How do you know?
Problem 5
It’s hot in Mathville. After buying your souvenir, you head over to a water balloon contest to cool off. You win the contest if you can fill up a tank exactly to the line: $200\oz$.
You have some water balloons of all different sizes. You can fill a balloon at the tap and toss it into the tank, or you could instead fill the balloon from the tank to get rid of some of the water. Each balloon can only be filled once, since it pops after you throw it. You have one $1\oz$ balloon, one $3\oz$ balloon, one $9\oz$ balloon, one $27\oz$ balloon, and so on for all the powers of $3$ up to infinity. (Mathville takes their water balloons very seriously.) If the tank starts empty, how can you use the balloons to get exactly $200\oz$ in the tank?
Problem 6
After winning that contest, you make it to the semifinals. The rules are the same, but the line has moved so that you have to fill the tank with a different volume. Is there any volume they could choose so that there is more than one way to fill the tank to that exact volume? How do you know?
Problem 7
In the finals for the contest, they introduce fractions. Instead of having bigger and bigger water balloons, they have smaller and smaller ones! You have one \qty{1}{oz} balloon, one $\frac13\oz$ balloon, one $\frac19\oz$ balloon, one $\frac1{27}\oz$ balloon, and so on. How many ways can you find to get exactly $\frac12\oz$ into the tank?
