Qualifying Quiz

## FAQ Answers

Many people ask similar questions about the statements of Quiz problems via our QQ helpline. As we receive questions, we will keep adding our answers here; you might find them helpful!

### Problem 1

Does the answer need to be an explicit formula?

It doesn't have to be an explicit formula; you could give a procedure instead. As long as your answer clearly describes what the shortest sequence would be for any n, it's good!

### Problem 2

In part (c), what do you mean by "describe the trajectory that the ball takes"? What information should an answer include?

We don’t have any specific requirements for the form of your description of the trajectory. From your description, we should be able to tell that you’ve figured out anything interesting there is to figure out about what the ball does (and proven it).

Was there a typo in the third rule?

Yes, it used to say "These two bounces are reversible: if the ball is moving up and left along a line with slope −1/2 or −1, it bounces off and continues moving down and right along a line with slope 1 or slope 2, respectively."

The correct phrasing is "These two bounces are reversible: if the ball is moving up and left along a line with slope −1/2 or −1, it bounces off and continues moving down and left along a line with slope 1 or slope 2, respectively."

### Problem 3

Can I keep track of which coins I've used in previous weighings?

Yes! For example, you can number the coins with a marker if you want, and write down which sets of coins you've included in each weighing. The "cannot distinguish" part just means you can't manually tell which have weight 1 or 2.

Do I have to know that a specific set of coins weighs exactly X grams, or do I just have to put a set weighing exactly X on the scale at some point?

You must be able to identify a set of coins that you can prove has the correct weight; it's not enough to just happen to weigh such a set at some point.

### Problem 4

In part (a), do both keys have to be in the starting room?

No; they could be, but you could also pick up one key and use it to get to another key in a different room.

In part (b), does the second key have to be somewhere you can get to only after picking up the first key?

No; it could be, but it could also be in the starting room.

Wait, then how are parts (a) and (b) different?

The difference is subtle! In part (a), you identify which key assignments are valid and choose one uniformly at random. In part (b), you place one key at a time, making sure that it can be reached using the keys that you have so far. Consequently, in part (b), different valid assignments may have different probabilities of being chosen.

How do I decide which chest in the starting room to open first? Is it random?

Because the keys are assigned randomly, it does not matter how you pick which chest to open first.

Does "picking up the Red Key first" mean "picking up the Red Key before any other keys" or "picking up the Red Key from the first chest you open?"

From the first chest you open. We've edited the Quiz to clarify this.

For part (c), what if a map has more keys than treasure chests?

If there are more keys than chests, then there are no valid assignments of keys to chests, so the probabilities are undefined. To satisfy part (c), your answer would need to be a map with at least one valid assignment--in particular, it would need at least as many chests as keys.

In part (b), is the second key placed after I pick up the first key (i.e. after some chests have already been opened)?

No; the key locations are all randomized before you start exploring the maze. When it says "after picking up the first key," it just means that the second key is placed somewhere that you will be able to get to once you have picked up the first key.

### Problem 5

Can players only move to numbers which are integers?

Yes, all positions need to be integers.

Does a pair of positions occur every time either player moves, or only after they have both moved?

Whenever either player moves, that counts as a pair of positions. For example, if Narmada moves to 3 on her first move, Travis moves to 5, Narmada moves to 4, and Travis moves back to n, then all of the positions (1,n), (3,n), (3,5), (4,5), (4,n) have occurred. Narmada is not allowed to move to 1 or 3 on her next move because it would create a repetition.