Solution to Bounce House

At the start of the puzzle, we are given three grids, with no instructions of how to fill them, although we are given the results of extracting the first two grids.

The title, as well as the fact that we can see pairs of reversible strings (such as PME/EMP, AM/MA, and LS/SL), suggests placing mirrors in the grid.

For the first puzzle, there's only one way to put mirrors such that the E sees the other E, and likewise the X. (Note that if we assume that spaces with mirrors cannot have letters, then this example cannot be satisfied. The E on the E path will necessarily be on the X path.

Thus, we can fill in the grid as follows:

	X	E
E		E
	X		X

This also suggests that extraction is done by looking at the letters on the mirrors, in row-major, standard reading order (i.e. reading left to right, top to bottom).

The first puzzle didn't necessarily need mirrors; if we thought the letters just denoted the letters "seen" when viewing in a certain direction, it would also work as well.

However, that interpretation doesn't work as well for the second grid, while the mirrors do. We can interpret the multiple-letter clues as the order of the mirror letters that we traverse along the path of the beam, explaining the string reversal, since going through the path from the other side will necessarily hit the same mirrors in reverse. (One additional observation that can be useful to note is that strings of even length are always on the same or opposite sides, while strings of odd length are on sides that are 90 degrees apart. This is because each mirror bounce converts the light beam from horizontal to vertical and vice versa, so parity dictates this.)

One compelling way to put the mirrors in the second grid such that the extraction works (reading out SAMPLE in row-major order) is as follows:

			MA	PME
SL	S	A	M	P
	L				LS
			E		EMP
		AM

We might notice that there are six regions as well as six letters in SAMPLE. This suggests that we want to put one mirror in each region, no more and no less.

(Without this constraint, there are multiple solutions for this puzzle. For example, the three AMP mirrors could be moved one cell higher.)

In this example, the mirror labeled M is used twice (in both MA/AM and PME/EMP). This suggests that mirrors have the same letter on both sides.

Now, it's time to solve the real puzzle! First, we'll need to solve the clues. Some of these clues are broken up into multiple parts; the answers to each part should be concatenated together. The use of the mirror constraint can also be helpful to answer the clues.

Clue	Answer
1. Fashionable	IN
2. Exclamation by a knight	NI
3. Saharan desert pavement	REG
4a. It's one hour ahead of UTC	CET (Central European time)
4b. Solid water	ICE
4c. Approx. a certain time	C (circa)
5. Kind of ID tag	RF (radio frequency, like an RFID tag)
6. Symbol for element 103	LR (lawrencium)
7a. ____ n'est pas une pipe	CECI
7b. Toshiba subsidiary making retail products, among others	TEC
8a. Location of westernmost U.S. Mint	SF (San Francisco)
8b. Death Note detective	L
9a. Avg. dist. from the Earth to the Sun	AU (astronomical unit)
9b. Any one of three basic skills taught in schools?	R (as in the 3 R's)
10. Language code of the first part of clue 7	FR (French)
11. Two in Maori	RUA
12a. ASCII 0x0A	LF (line feed)
12b. Letter used to denote entropy	S
13. Former "do" in music	UT
14. Mongolian yurt	GER
15. Et ____, brute?	TU
16. Area of ML involving maximizing a reward function	RL (reinforcement learning)

Armed with the clue answers, we can solve the grid as follows. Note that one of the paths uses a mirror twice; indeed that letter counts twice. A full logical solution follows in the appendix.

			1		2	3	4	5
16						R			E
				F		L				6
	E						C			7
15	T		I		N
14									G
				S						8
13	U							R	F
	A									9
			C				E
						12		11	10

Reading the mirrors in row-major order gives the cluephrase REFLECTING SURFACE, which clues the answer MIRROR.

Appendix: Solving the Logic Puzzle

First, we can pair up the clues as follows, noting down the lengths of the strings.

The B clue must have two mirrors in rows 1 and 2. The only place this can happen without violating one mirror per region is column 6. We can also mark the other squares on the path that don't have mirrors.

We can additionally mark a few spots in columns 3 and 5 where there can't be mirrors due to clue A, and mark the other spots in the regions of the two mirrors we placed earlier as empty.

We can do a similar argument for clue C. Rows 1–3 can't have the left mirror, and rows 8–9 can't have the right mirror.

Now, clue D must turn at R1C9, and thus the final mirror is at R5C9.

This lets us complete clue C as well.

Now, a slightly tricky step: clue A's right mirror must be on one of the two remaining spaces of the plus-shaped region in the middle. So the left mirror means that we can rule out four of the spaces in the Y-shaped region to the left of the plus-shaped region, as it must be on one of the other two spots.

We can place two mirrors in clue F now in the third row. The R3C1 mirror can't point upwards since there is no way to point the beam back right.

We can deal with clue G, partially, as well. The remaining two mirrors have to be in rows 7 and 8, ruling out the positions in columns 4–7.

This gives us another mirror for clue F.

Clue E is also fixed here.

This lets us set clue A's mirrors in place.

Here, we could use the letter constraint of the letters to deduce that clue H must be on column 1 and not column 3. Alternatively, we can rule out the other case with a small look-ahead. If clue H's mirrors were on column 3, clue F must hit a mirror at R6C1, going right to R6C4 and then going down to R9C4. However, clue G must have a mirror at R8C3, violating the one mirror per region constraint.

At this point we can satisfy clue G...

...and finally clue F as well, completing the puzzle!