Eclectic Spatial Geometry (Solution)

#### by Bryce Herdt, with assistance from Timothy Chow

The "home plane" referred to in the clues is the mostly-planar mural located at the ESG building on campus. Solvers need to see that mural or a facsimile to confirm the geckos' adjacency and colors. Just in case solvers are nervous about approaching the Manic Sages' headquarters, though, the eleven clues form an acrostic for SEE THE MURAL.

From clue 3, the "seven geckos from across the middle of the plane" are in a row of four over a row of three. The "appropriate rock band" is named ESG (Emerald, Sapphire and Gold), and the fourteenth track on their album "A South Bronx Story" is GET FUNKY. The "similarly-named Southern rapper" is E.S.G., and the second track on his album "City under Siege" is RIDE OUT. (The letters M and C stand for Mathcamp, the other side of the Manic Sages team, and hold no secret that's important to solving the puzzle.) Here are most of the geckos' starting letters:

```  ?
/ \
R---I---D---E
\ / \ / \ /
O---U---T
\ /
?
```

To start filling in the faces, note that from the geckos' starting positions, the letters E, T, and U must be on the equator with T touching the other two. The equator faces are isomorphic, so since only the gecko from clue 10 cares about handedness, it doesn't matter yet where these three letters go.

Note that D, I, O, and R must lie in the same hemisphere. D must share an edge with T, but could go on either the square or the triangle. By clue 10, O has one letter at 15-o'clock and another, K, at 23-o'clock, so it's on a triangle. And U is adjacent to I, funnily enough. Here are two possibilities, as distinguished by which face type D is on:

```   *-----*            *-----*
/|     |\ U      U / \ O / \
/ |  I  |O\        /   \ /   \
*--*-----*--*      *  I  *     *
|  |     |  |      |\   / \   /|
|  |     |  |      | \ /   \ / |
|  |     |D | T  T |D *     *  |
|  |     |  |      | / \   / \ |
|  |     |  |      |/   \ /   \|
*--*-----*--*      *     *     *
\ |     | /        \   / \   /
\|     |/ E      E \ /   \ /
*-----*            *-----*
```

The green gecko in clue 6 travels a 16-face path (not all distinct) crossing 15 edges, although the six faces she visits that "count" aren't obvious yet. Still, ".HU.FQ.AW.EN.VE." gives several pointers. On just the equator, N touches E, and something shares edges with U and F. Since O and K still have to touch, that's three more letters placed in each possible case.

```      K                  K
*-----*            *-----*
F /|     |\ U      U / \ O / \ F
/ |  I  |O\        /   \ /   \
*--*-----*--*      *  I  *     *
|  |     |  |      |\   / \   /|
|  |     |  |      | \ /   \ / |
|  |     |D | T  T |D *     *  |
|  |     |  |      | / \   / \ |
|  |     |  |      |/   \ /   \|
*--*-----*--*      *     *     *
\ |     | /        \   / \   /
\|     |/ E      E \ /   \ /
*-----*            *-----*
N                  N
```

Actually, clue 8 says that two geckos must step from I and U onto F and J in some order, so the possibility shown on the right can be eliminated. Then another couple of letters can be placed: R by adjacency, J by clue 8, and H by clue 6. And since either M or C must touch both T and U, while clue 8 says C touches F, M can also be placed.

```        K               K
*-----*         *-----*
F /|     |\ U   U / \ J / \ F
/ |  I  |O\     /   \ /   \
*--*-----*--*   *  H  *     *
|  |     |  |   |\   / \   /|
|  |     |  |   | \ /   \ / |
|  |  R  |D | T |M *     *  |
|  |     |  |   | / \   / \ |
|  |     |  |   |/   \ /   \|
*--*-----*--*   *     *     *
\ |     | /     \   / \   /
\|     |/ E   E \ /   \ /
*-----*         *-----*
N               N
```

It so happens that this is the correct orientation for the yellow gecko from clue 10. She might travel from O to either R or T, but she wouldn't be able to move off T. So instead, her first two moves are O to R and then directly to the left.

That square... could that hold C? It turns out no. Going 3-o'clock from C would take the yellow gecko to I, to D, to nowhere else. So the C is in the remaining triangle adjacent to I. Then since by clue 8, Y touches one of C and F, both Y and G can be placed.

```        K               K
*-----*         *-----*
F /|     |\ U   U / \ J / \ F
/C|  I  |O\     /   \ /   \
*--*-----*--*   *  H  *     *
|  |     |  |   |\   / \   /|
|  |     |  |   | \ /   \ / |
Y |  |  R  |D | T |M *     *  | Y
|  |     |  |   | / \   / \ |
|  |     |  |   |/   \ /   \|
*--*-----*--*   *     *     *
\ |     | /     \   / \   /
G \|     |/ E   E \ /   \ / G
*-----*         *-----*
N               N
```

To be sure, that square between R and Y has to hold one of CFILORUX. But with CFIORU taken, and X directing to C without the possibility of a fourth move, it must be L. Then the indicated triangle must contain one of DHLPTX, and with DHLT taken and X going back to L, it contains P. What the heck, add Q next to F as per clue 6, along with A, W, and V.

```        K               K
*-----*         *-----*
F /|     |\ U   U / \ J / \ F
/C|  I  |O\     /   \ /   \
*--*-----*--*   *  H  *  Q  *
|  |     |  |   |\   / \   /|
|  |     |  |   | \ /   \ / |
Y | L|  R  |D | T |M *  W  *  | Y
|  |     |  |   | / \   / \ |
|  |     |  |   |/   \ /   \|
*--*-----*--*   *     *  A  *
\P|     |V/     \   / \   /
G \|     |/ E   E \ /   \ / G
*-----*         *-----*
N               N
```

Clue 8 puts B in the last empty space next to Y. The gecko in clue 9 is the one who started on C, so X is on the square between W and E. By that clue, S and M are in the same hemisphere, so that finally disambiguates S and Z.

```        K               K
*-----*         *-----*
F /|     |\ U   U / \ J / \ F
/C|  I  |O\     /   \ /   \
*--*-----*--*   *  H  *  Q  *
|  |     |  |   |\   / \   /|
|  |     |  |   | \ /   \ / |
Y | L|  R  |D | T |M *  W  * B| Y
|  |     |  |   | / \   / \ |
|  |     |  |   |/   \ /   \|
*--*-----*--*   *  X  *  A  *
\P|  Z  |V/     \   / \   /
G \|     |/ E   E \ / S \ / G
*-----*         *-----*
N               N
```

Well, then. Now to plot the geckos' paths. Here's what has been tracked so far:

```  4   5   6   7    8   9   10  11
+---+---+---+---+----+---+---+---+
| R |   | M | I | TD | C | O |   |
+---+---+---+---+----+---+---+---+
|   |   | K |   | UI | Y | R |   |
+---+---+---+---+----+---+---+---+
|   |   | B |   | JF | Q | L |   |
+---+---+---+---+----+---+---+---+
|   |   | X |   | FC | K | P |   |
+---+---+---+---+----+---+---+---+
|   |   | Z |   | BY | I | G |   |
+---+---+---+---+----+---+---+---+
|   |   | T |   | YP | L | Z |   |
+---+---+---+---+----+---+---+---+
```

The two remaining geckos start on E and U. The one in clue 11 can't be from U, since U is right next to H. So she's the gecko from E, and the path is then trivial to plot. (It could have been done anyway, without knowing anything about the starting point, but it's easier now.)

```  4   5   6   7    8   9   10  11
+---+---+---+---+----+---+---+---+
| R | U | M | I | TD | C | O | E |
+---+---+---+---+----+---+---+---+
|   |   | K |   | UI | Y | R | M |
+---+---+---+---+----+---+---+---+
|   |   | B |   | JF | Q | L | W |
+---+---+---+---+----+---+---+---+
|   |   | X |   | FC | K | P | A |
+---+---+---+---+----+---+---+---+
|   |   | Z |   | BY | I | G | Q |
+---+---+---+---+----+---+---+---+
|   |   | T |   | YP | L | Z | J |
+---+---+---+---+----+---+---+---+
```

The gecko in clue 6 can't go to F and immediately to S, so that path is also fairly easy to plot. So is the path starting from R. (We'll come back to that in a second.)

```  4   5   6   7    8   9   10  11
+---+---+---+---+----+---+---+---+
| R | U | M | I | TD | C | O | E |
+---+---+---+---+----+---+---+---+
| D |   | K | L | UI | Y | R | M |
+---+---+---+---+----+---+---+---+
| E |   | B | Y | JF | Q | L | W |
+---+---+---+---+----+---+---+---+
| S |   | X | Q | FC | K | P | A |
+---+---+---+---+----+---+---+---+
| A |   | Z | H | BY | I | G | Q |
+---+---+---+---+----+---+---+---+
| W |   | T | U | YP | L | Z | J |
+---+---+---+---+----+---+---+---+
```

Clue 5 is the fiddliest. From U, the gecko can go to either T or O, since M's occupied. From T, he can only reach M, but then the rest of the path could only be FFV. So his first move is to O, and the rest of the path is unambiguous.

```  4   5   6   7    8   9   10  11
+---+---+---+---+----+---+---+---+
| R | U | M | I | TD | C | O | E |
+---+---+---+---+----+---+---+---+
| D | O | K | L | UI | Y | R | M |
+---+---+---+---+----+---+---+---+
| E | I | B | Y | JF | Q | L | W |
+---+---+---+---+----+---+---+---+
| S | D | X | Q | FC | K | P | A |
+---+---+---+---+----+---+---+---+
| A | R | Z | H | BY | I | G | Q |
+---+---+---+---+----+---+---+---+
| W | V | T | U | YP | L | Z | J |
+---+---+---+---+----+---+---+---+
```

Now, the correct order of the five words in clue 4 is MALLARD IDEOGRAM NICEST UNSAVORY SCOFFLAW, and this happens to be another acrostic, for the hint MINUS. Even without this hint, solvers may note that each gecko's favorite letter comes later in the alphabet than their starting letter. Taking cues from clues 7 and 10, the nine differences, in clue order, spell the answer EAGLELIKE.