Solution to the Characterisation puzzle

Characterisation is one of the interesting puzzles of the Sydney University Puzzle Hunt 2010 competition. The plot of the game that year was based on Lewis Carroll's "Alice's Adventures in Wonderland" and "Through the Looking-Glass and what Alice found there". The game consisted of numerous scenes that were improvisations on the famous "A Mad Tea-Party". Each scene began with a short narrative that provided context for the task and could also contain some hints. Characterization was the last puzzle of the game before the final Meta Puzzle and was rated five stars out of five in difficulty.

These biscuits aren’t sweet enough. Pass the sugar, would you?” asked the Mad Hatter, and the March Hare pushed the pot over to him obligingly. He poured the sugar over the plate of biscuits as he continued.

“The Red Queen’s fixed, furious gaze faltered and tore from Alice’s back, for her nemesis was preparing to strike. Losing all composure, she turned to face the White Queen. ‘She is mine, and so is this place, sister!’ she cried as she lunged towards the White Queen, but stopped just short, realising what an error she had made. That little girl, the cause of so much strife, was now beyond her reach! The Old White King and the White Queen prepared to strike, as the brave knight stood by, powerless to stop them.”

The March Hare began dozing on his stool, and then began snoring loudly, his jowls flapping in his fitful rest.

“Hey, lazyface!” said the Dormouse, throwing the teapot at the rabbit’s head. It hit with a clang and soared into the air, and the March Hare sat up, wide-eyed with surprise, just in time for it to strike him again as it came back to earth. Alice and the Mad Hatter laughed as the two creatures shot furious glances at each other over the table.

“Now now, my guests, we wouldn’t want Alice to think we’re mad, would we?” said the Mad Hatter, and though he sounded serious, it made Alice laugh even harder.

“Let’s see if she’s mad like us,” said the Dormouse. “Give it to her, Hare!”

The March Hare, now somewhat recovered, leapt beneath the table and retrieved a glimmering, colourful cube. He bounced it off the tabletop, and it landed perfectly in Alice’s lap. What on earth could this be?

After the end of the game the organizers provided a detailed solution to the task.

Solution

This puzzle initially seems to present a lot of information at once. We have five 5x5 square grids suspended on different levels or layers, tokens placed on these grids that come in a variety of shapes and colours, and blue and red lines drawn between different squares on each level. Of the token types, we have five different colours (red, yellow, green, blue, purple) and five different shapes (hearts, triangles, diamonds, spades, clubs), or 25 different kinds of tokens total. It is therefore reasonable to assume that we want to arrange these 25 different tokens somehow on each level, given each level has exactly 25 squares in its grid, and duplicate tokens appear in the puzzle on different layers.

The next step then is to determine how exactly these 25 tokens should be arranged on the grids per level. The only information we have not yet used is the supplied red and blue lines. Looking at the top layer for help seems reasonable here since we are given five tokens on its grid, as opposed to the four supplied on every other level. Interestingly, the central token, a green spade, is connected by a blue line to the green heart, and by a red line to the yellow spade.

The suggestion here is that a blue line is drawn between any two adjacent (including diagonally adjacent) squares whose tokens have the same colour, while a red line is similarly drawn between any two adjacent squares whose tokens have the same shape. This idea is supported by the fact no blue and red lines occupy the same space (or else the adjacent squares in question would have to host identical tokens) and that when at any point a three-square L-shape is formed by lines of the same colour, its endpoints are also joined by that colour's line (since the endpoints are diagonally adjacent squares whose tokens must share the same colour/shape). Finally, the puzzle title, Characterisation, hints at us having to consider the characteristics of the various tokens.

If we then follow these rules, it is possible to determine a unique configuration of the 25 tokens for each layer, as illustrated below (see the Addendum for general tips on how to derive the filled-out layers [followed by example solutions]):

Thus far we have not used the positioning of these layers in solving any steps of the puzzle, so this fact seems like the best thing to consider next. The story text describes the levels as being contained with a cube, and similarly the perspective of the puzzle image suggests we want to imagine the layers to lie directly above/below each other. The logical step to take from here is to consider other slices of the cube - that is, to look at the 5x5 grids formed by slicing the cube in layers perpendicular to the levels we were previously working on.

Doing so for the "side" slices of the cube (i.e. those slices facing left/right when looking at the "cube" head-on) gives the following new grids, going from left to right:

It should be fairly clear here that a letter is formed in each slice by a different colour; this is promising, since the title Characterisation somewhat hints at this (we are forming characters, or letters, by singling out certain characteristics). In fact, going from left to right, the letter is formed by the colours red, yellow, green, blue, and purple respectively - the natural rainbow colour ordering. Furthermore the letters we get are R, G, P, Y, B, which also happen to be the first letters of the colours used in the puzzle (red, green, purple, yellow, blue).

However an arbitrary colour ordering does not seem to be enough to constitute an answer or an indication as to the next step. The next thing to do is consider those grids found by taking slices of the cube along its third axis, i.e. by considering those slices facing forward/back. From the front to the back slice, we get:

It follows that from this different perspective, the characters should be formed by tokens of the same shape rather than colour. Although some of the letters here are less obvious, they should still be easily identifiable provided we can use some natural ordering of the different types of shape (just as the colours were ordered by rainbow order). The most obvious ordering is by increasing side/vertex number, namely heart (2 sides/corners), triangle (3), diamond (4), spade (5), club (6). By looking at the hearts in the first slice, the triangles in the second, etc., we obtain the letters S, D, C, T, H, which again correspond to the shape names (spades, diamonds, clubs, triangles, hearts).

So now we have derived an ordered list of colours and an ordered list of shapes. Separately, these lists don't seem particularly helpful, but together they naturally suggest five particular tokens: the red spade, green diamond, purple club, yellow triangle, and blue heart. The final step then is to figure out what to do with this ordering of five particular tokens. Returning to the original grids (those with all 25 tokens per grid), the most logical thing to do is draw lines between the tokens in the order given (i.e. red spade - green diamond - purple club - yellow triangle - blue heart). This gives the following (from top layer to bottom):

We have clearly drawn yet another set of characters, this time spelling out the word Rebus, which is itself a term meaning pictorial representations of words or characters. [This is the answer to the task.]

Addendum

There are many ways to deduce the positions of each token on each of the five grids/layers, but some of the more common techniques are listed here:

• Keep a list of all 25 possible tokens and mark off those that have already been used; once you know you have used up all instances of a particular shape or colour, the filling-out puzzle becomes a lot easier.
• Once four tokens of the same shape/colour have been identified, the fifth token of that shape/colour is also immediately determined.
• Very often the absence of a red or blue line can be just as if not more useful than the presence of one, since processes of elimination can be used to ultimately determine what shape/colour lies in these isolated squares.
• For any single blue or red line running over n consecutive squares, call it an n-line (for instance the first layer has one blue 5-line, two blue 4-lines, and three blue 2-lines). For each colour/shape, the n-lines they lie on must total 5, where lying on a completely disconnected (by blue/red lines respectively) square counts as 1 to their total. So if we know, for instance, that in the first layer, neither of the blue 4-lines nor the blue 5-line hold blue tokens, then the blue tokens can only lie on two 2-lines and one isolated square, one 2-line and three isolated squares, or five isolated squares. Combining this technique with the previous one is often very powerful.

Examples of filling out the layers

First layer

1. (2, 1) = spade/?; (3, 2) = spade/?.
2. (4, 1) = heart/?.
3. (4, 4) = club/?; (3, 5) = club/?.
4. (3, 4) = ?/green; (2, 5) = ?/green; (4, 2) = ?/green.
5. (4, 2) = club/green; (3, 1) = club/?.
6. (3, 4) = triangle/green; (2, 3) = triangle/?; (2, 2) = triangle/?; (1, 2) = triangle/?.
7. (2, 5) = diamond/green; (2, 4) = diamond/?; (1, 4) = diamond/?.
8. (1, 3) = heart/?.
9. (5, 2) = spade/?.
10. (2, 1) = spade/red; (1, 2) = triangle/red; (1, 3) = heart/red; (1, 4) = diamond/red.
11. (2, 2) = triangle/purple.
12. (2, 3) = triangle/yellow; (2, 4) = diamond/yellow.
13. (3, 2) = spade/blue; (4, 1) = heart/blue.
14. (3, 1) = club/yellow.
15. (5, 2) = spade/purple; (5, 3) = ?/purple; (5, 4) = ?/purple; (4, 4) = club/purple.
16. (3, 5) = club/blue; (4, 5) = ?/blue.
17. (4, 3) = ?/yellow.
18. (4, 3) = heart/yellow; (5, 3) = heart/purple.
19. (4, 5) = diamond/blue; (5, 4) = diamond/purple.

Second layer

1. (1, 3) = triangle/?; (2, 1) = triangle/?; (3, 2) = triangle/?.
2. (1, 4) = heart/?.
3. (4, 2) = club/?.
4. (4, 4) = spade/?; (3, 4) = spade/?; (2, 3) = spade/?.
5. (1, 5) = ?/yellow.
6. (4, 3) = ?/purple; (3, 3) = ?/purple.
7. (3, 3) = heart/purple; (2, 2) = heart/?; (3, 1) = heart/?.
8. (4, 3) = diamond/purple; (5, 3) = diamond/?.
9. (3, 2) = triangle/green; (2, 2) = heart/green.
10. (2, 3) = spade/red; (1, 4) = heart/red.
11. (3, 1) = heart/blue; (4, 2) = club/blue.
12. (2, 1) = triangle/red; (1, 1) = ?/red.
13. (1, 1) = diamond/red.
14. (2, 4) = diamond/?; (1, 5) = diamond/yellow.
15. (3, 5) = club/?; (4, 5) = club/?; (5, 5) = club/?.
16. (3, 5) = club/purple.
17. (5, 3) = diamond/green.
18. (2, 4) = diamond/blue.
19. (4, 4) = spade/blue.
20. (3, 4) = spade/green; (4, 5) = club/green.
21. (5, 5) = club/yellow.
22. (5, 1) = triangle/purple.
23. (5, 2) = spade/yellow.
24. (1, 3) = triangle/yellow.

Third layer

1. (2, 4) = diamond/?; (3, 3) = diamond/?; (2, 2) = diamond/?.
2. (3, 1) = club/?; (4, 2) = club/?.
3. (4, 3) = spade/?; (4, 4) = spade/?.
4. (1, 3) = ?/red; (1, 2) = ?/red.
5. (3, 2) = ?/blue; (4, 3) = spade/blue.
6. (5, 5) = ?/green; (4, 4) = spade/green; (3, 4) = ?/green; (3, 3) = diamond/green.
7. (5, 3) = ?/purple; (5, 4) = ?/purple.
8. (5, 5) = club/green.
9. (3, 4) = heart/green; (2, 5) = heart/?; (1, 5) = heart/?.
10. (3, 5) = club/?.
11. (2, 3) = ?/yellow; (2, 2) = diamond/yellow, (1, 1) = ?/yellow.
12. (5, 1) = ?/blue.
13. (4, 1) = ?/red; (4, 2) = club/red.
14. (3, 1) = club/purple.
15. (3, 5) = club/yellow; (2, 5) = heart/yellow.
16. (4, 1) = heart/red.
17. (3, 2) = triangle/blue.
18. (5, 1) = diamond/blue.
19. (2, 4) = diamond/purple.
20. (1, 5) = heart/blue.
21. (5, 3) = triangle/purple.
22. (5, 4) = heart/purple.
23. (1, 1) = triangle/yellow; (1, 2) = triangle/red.
24. (1, 3) = spade/red; (2, 3) = spade/yellow.

Fourth layer

1. (1, 2) = triangle/?; (2, 3) = triangle/?; (3, 2) = triangle/?.
2. (2, 2) = club/?; (1, 1) = club/?.
3. (2, 5) = diamond/?.
4. (4, 3) = spade/?; (4, 4) = spade/?; (5, 5) = spade/?.
5. (2, 4) = ?/blue; (1, 5) = ?/blue; (4, 4) = spade/blue.
6. (4, 2) = ?/yellow.
7. (3, 2) = triangle/blue.
8. (2, 5) = diamond/yellow.
9. (2, 1) = diamond/?.
10. (1, 2) = triangle/yellow; (2, 2) = club/yellow.
11. (4, 2) = heart/yellow; (5, 1) = heart/?.
12. (4, 1) = spade/?.
13. (3, 3) = diamond/?.
14. (5, 2) = diamond/?.
15. (3, 3) = diamond/purple; (4, 3) = spade/purple; (5, 4) = ?/purple.
16. (5, 1) = heart/purple.
17. (5, 4) = triangle/purple.
18. (4, 5) = heart/?.
19. (1, 4) = heart/?; (1, 5) = heart/blue.
20. (2, 4) = club/blue; (3, 4) = club/?.
21. (2, 3) = triangle/red; (1, 4) = heart/red.
22. (3, 4) = club/green; (4, 5) = heart/green.
23. (5, 5) = spade/red.
24. (1, 1) = club/red; (2, 1) = diamond/red.
25. (4, 1) = spade/green; (5, 2) = diamond/green.

Fifth layer

1. (1, 2) = triangle/?; (2, 2) = triangle/?.
2. (4, 2) = diamond/?; (4, 3) = diamond/?; (5, 3) = diamond/?.
3. (4, 5) = heart/?.
4. (2, 1) = ?/red; (3, 1) = ?/red.
5. (2, 5) = ?/blue.
6. (4, 1) = ?/yellow; (3, 2) = ?/yellow; (2, 3) = ?/yellow; (2, 4) = ?/yellow.
7. (4, 4) = ?/green; (3, 4) = ?/green.
8. (2, 4) = triangle/yellow.
9. (5, 4) = triangle/?.
10. (3, 5) = spade/?; (4, 4) = spade/green; (2, 1) = spade/red; (2, 3) = spade/yellow.
11. (3, 2) = club/yellow; (3, 3) = club/?; (3, 4) = club/green.
12. (1, 3) = club/?; (1, 4) = club/?.
13. (2, 5) = diamond/blue.
14. (3, 1) = heart/red; (4, 1) = heart/yellow; (5, 2) = heart/?.
15. (3, 5) = spade/purple.
16. (4, 5) = heart/blue.
17. (5, 2) = heart/purple; (5, 3) = diamond/purple; (5, 4) = triangle/purple.
18. (4, 3) = diamond/red.
19. (4, 2) = diamond/green.
20. (1, 2) = triangle/green.
21. (2, 2) = triangle/blue; (1, 3) = club/blue.
22. (3, 3) = club/purple.
23. (1, 4) = club/red.