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 Mind Boggling (Posted on 2003-06-11)
In the popular game of Boggle, each of 16 dice has six different letters (QU substitutes for a single letter on one cube face) on it. The dice are shaken and fall into a 4x4 square, so that one letter on each cube shows. Players have three minutes to form as many words of length three or longer by moving from letter to adjacent letter either vertically, horizontally, or diagonally. In this game, players may return to a letter in forming a word but may not pause on a letter. Given the sample board:
``` F B U P
T M E O
R S H I
E T B W
```
possible words are POEM, PESTER, HOPE, THEME, and RESETS, but not MESS.

In a recent game of Boggle, Chad and his dad formed the words given in the clues below, as well as many others. The following observations were made:
1. Both Chad and his dad got the words AGE and PINT, each of which showed up on the board in a straight line.
2. Chad scored big points by making FRAGMENT and PALACE, neither of which his father had on this final word list.
3. Dad scored points with CHORAL, which Chad didn't see. Both players anagrammed FAINT and ARCHLY from the board.
4. Both formed COMET, starting with a horizontal move from C to O.
5. The letter W was in the upper lefthand corner of the 4x4 square.
Reconstruct the arrangement of the top dice letters that showed on their game board.

(puzzle originally from www.allstarpuzzles.com)

 See The Solution Submitted by DJ Rating: 4.1429 (21 votes)

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 solution | Comment 1 of 8
There are 16 different letters represented in the words, so each of the different letters must be present exactly once in the grid. The letter A is used twice in one word and so must be returned to in forming that word (PALACE).

The letter A is adjacent to 7 different letters (L, C, G, I, P, F and R) in the various words. Therefore the A must be in one of the central 4 positions, so that it has 8 neighbors-the 7 that it needs plus exactly one that it doesn't need.

Since the letter A needs to be adjacent to P and I, and the letter E must be adjacent to N (because of FRAGMENT) and T (because of COMET), the straight-line words AGE and PINT must be parallel and adjacent to one another and going in the same direction.

AGE and PINT can't be diagonal, as PINT would have to be in one of the two major diagonals in order to fit, and then AGE would be in an adjacent diagonal with only 3 letters; but that would place A outside the central square of 4 and is thus no good.

Thus AGE and PINT must be both either vertical or horizontal, and with A not on an edge, so the A is orthogonally connected to the I and diagonally connected to P and N, and then further, the N is the only letter adjacent to A whose adjacency is not really needed, so any future letters placed next to A must be from the remainder of the required set (L, C, F and R).

The C must be next to the A and the E (from PALACE), and so must be orthogonally adjacent to the G in AGE (obviously the other side from the N in PINT). The O in COMET is horizontally connected to the C, but cannot be next to the A as it's not one of the remaining 4 to be connected to A, and it cannot be next to the E, as that would leave no room for the M between the O and the E. Therefore AGE and PINT must be vertical rather than horizontal so that CO can proceed horizontally away from this pair, as in the following diagram:
```
P

AI

OCGN

ET

```

Note that this diagram may be reversed left-right, or up-down, as we don't yet know where the optional space will come about for the W. At the end, we can adjust by flipping left-right or reversing the order of the rows to place the W in the upper left corner of the final result.

The M in COMET must be placed in the only position adjacent to both the O and the E. The R in ARCHLY must then be in the only position next to both the A and the C. The H in archly must then be placed above the O in the above diagram as it must be adjacent to the C, but going below the O would not allow enough room to complete ARCHLY (and the L must be next to the A anyway). That being said, the L must go in the only position left that's adjacent to the O and the A. The Y must go to the top left, leaving room for the F to go next to the A by the P.

That leaves us with:
```
YLFP

HRAI

OCGN

MET

```

So the only place for the W on this diagram is the lower left, which we get to the upper left by reversing the order of the rows:

```
WMET

OCGN

HRAI

YLFP

```

 Posted by Charlie on 2003-06-11 02:58:55

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