# The Cafe

### by Alex Pei, Christopher Reyes, Edgar Chen, Katie "chuttiekang" Chung, and Tracey Lin

This is the metapuzzle for the Atrium round.

We are presented with a menu with pictures depicting various food items. Based on the suspicious menu item names, we realize that each menu item strongly hints at a food chain. For example, “Greens with Sweet Dressing” references Sweetgreen. The food chains’ names are in alphabetical order, and the food items colored with the chain’s logo colors, which gives further confirmation.

The next observation is that the feeder answers have one word in common with at least one item on one food chain’s menu. For example, Sweetgreen’s menu includes the Harvest Bowl, which matches with the feeder “HARVEST FESTIVAL.” This allows us to match feeders to menu items as follows:

DonutsDunkin’Marble Frosted DonutMARBLE
PancakesIHOPSplit Decision BreakfastRANDOM DECISION
Jujube JuiceJambaIsland PitayaVINDICATION ISLAND
JJ’s SandwichJimmy John’sTotally TunaTOTALLY UNACCEPTABLE
Southern Fried ChickenKFCA La Carte WingHIND WING
Mama Bear’s Instant NoodlesPanda ExpressBeijing BeefBEIJING TIGERS
Greens with Sweet DressingSweetgreenHarvest BowlHARVEST FESTIVAL

The next step is to solve the logic puzzle. We see that our feeder answers give us 112 letters and we have a grid with 112 boxes, so we want to fill in the grid with our feeder answers. This allows us to know how far each person must go to get to a seat – for example, the person carrying Donuts wants to walk 6 spaces to fill in the word MARBLE. Here is the completed grid (colors for distinguishing people’s paths):

We take the letters from the squares marked with the numbers 1 through 19 to get the answer COUNTRY FRIED MISTAKE.

### Example Solve Path:

The first thing we take note of is the feeders’ parity, meaning what feeder lengths can get to what chairs. For example, even though HIND WING is very close to the center left chair, it cannot get to it because it would need to travel an odd-length path, but HIND WING is an even length answer. From parity, we note that only CARBON SINK, HARVEST FESTIVAL, and TOTALLY UNACCEPTABLE can reach the center left, center top, and center right chairs. CARBON SINK is so short that the only chair it can reach is the center left chair. (Note that in the below picture, we’ve cleaned up the grid by removing all the numbers from 1-19 and replacing the peoples’ images with the length of their feeder).

From CARBON SINK’s placement, we know that the path going to the bottom-left-most chair must go down and right below CARBON SINK. We can also claim some of the squares right below the CARBON SINK chair for the CARBON SINK path, as this is the only path that can reach these squares.

There are now only two chairs that the HIND WING path can get to – the two top left chairs. If we assume that the chair in the upper row, we find that (1) we’d need to cover all the area to the left of the chair by a single path as there’s only one entrance to this area and (2) this is too much area for any of our paths to cover, as seen below:

This means that HIND WING must go to the top chair of the left column.

The next chair we can match is the ROAD TO HELL chair. The ROAD TO HELL path can get to the top left chair as well as the two top right chairs. However, if the ROAD TO HELL path goes to the top right chairs, we effectively create an area with more chairs than people. In the below example, we have two chairs that are only accessible by the BEIJING TIGERS path.

This means that ROAD TO HELL goes to the top left chair. The top left four squares must either belong to the ROAD TO HELL path or the HIND WING path as there is only one entrance into these squares. However, HIND WING is not long enough, so these belong to the ROAD TO HELL path.

Now, the square between the ROAD TO HELL and HIND WING person must belong to one of these two paths. But now the ROAD TO HELL path is no longer long enough to get here, so the HIND WING path must take it.

Only CARBON SINK and HIND WING can reach the square above the CARBON SINK chair. However, if we assume CARBON SINK takes it, no path can get to the square to the left of the HIND WING person.

This means that HIND WING must take this square. We can fill in a few more squares in this area that the HIND WING path must take as a result. This also means that the ROAD TO HELL path must take the squares to the right of the HIND WING chair, as it’s the only path that can reach these squares. The square to the left of the HIND WING person must also be taken by CARBON SINK as it’s the only path that can reach it.

There is only one way for CARBON SINK to take all these squares, and it’s by taking the square right of the RANDOM DECISION. This also forces HIND WING’s and CARBON SINK’s paths as a result.

The RANDOM DECISION path cannot go down to take the bottom left chair’s path, and the HARVEST FESTIVAL cannot take this path because of parity reasons. This means that the bottom left chair’s path continues to the right as follows:

The HARVEST FESTIVAL path is not long enough to get to the middle top chair, which means it must go to the center right chair. Additionally, this path needs to go above the MARBLE path, else we’ll make an area with two chairs and a single person (VINDICATION ISLAND). However, it cannot take the squares two to the left of the MARBLE person, else the RANDOM DECISION person will have no chair to go to. This means we can fill the following:

From this, there is exactly one path MARBLE can take to get to a chair.

This also means that VINDICATION ISLAND must take the bottom right four squares as well as the path to the bottom left chair. With these two constraints, we can fill out the rest of the VINDICATION ISLAND path as there’s only one possibility.

RANDOM DECISION is barely long enough to get to the top left two chairs. This means it cannot fill any extra squares in the bottom area. This means that HARVEST FESTIVAL must take up the remaining space. Once it does so, it means we can fill out the rest of the HARVEST FESTIVAL path.

The RANDOM DECISION path must take the bottom right most squares or else it will cut off these squares from the remaining people. With this info, we can finalize the RANDOM DECISION path.

Due to parity, the BEIJING TIGERS path must take the rightmost chair. Due to this, it must also take the four top-right squares. Due to taking these top-right squares, it can no longer take the four squares directly right of the BEIJING TIGERS person, so these must belong to the TOTALLY UNACCEPTABLE path.

The BEIJING TIGERS path cannot take the bottom right most square as this path would create one square inaccessible from TOTALLY UNACCEPTABLE.

This means that TOTALLY UNACCEPTABLE must take the bottom right square. We can fill in the following squares immediately after that:

From this information, there is only one path that TOTALLY UNACCEPTABLE can take now, so we finalize it. And once TOTALLY UNACCEPTABLE is set, BEIJING TIGERS can be filled out, which finishes the whole grid.