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Another Game of Boggle
Can you solve the letter arrangement of the 4x4 game board?
October 3, 2005
From the introduction, words are made by moving from adjacent letter to adjacent letter vertically, horizontally, or diagonally; a letter may be returned to in a word, but the same letter can't be used twice in a row. The Boggle board in the game played ended up with 16 different letters: ABDFGHIKLNORSTWY. By clues 1 through 4, S touches ITAKWL, six different letters; so S must be one of the center four letters (the corner tiles are touched by three other tiles, while the other outer edge tiles are touched by 5). Similarly, I touches ATHRBSG (7), R touches IDOBGA (6), and A touches FISWYRN (7); so IRA fill the other three center positions of the board. By clue 3, since the L and Y must go in the outer ring of tiles, there are three possible arrangements for SLY:

   .L..   .L..   ....
   YS..   Y.S.   ....
   ....   ....   ..SL
   ....   ....   ..Y.

Trying the first arrangement, from SWAY in clue 2, the A, which is a center letter, must be below the S, with the W, which must touch both the S and A and be an outside letter, below the Y as follows:

   .L..
   YS..
   WA..
   ....

The T in SIGHTS (clue 2) is an outside letter and would be to the upper right of S. Then from ASK (2), the K would be in the upper-left corner:

   KLT.
   YS..
   WA..
   ....

Given the word FAITH (1), I would be to the right of S; R then would be the fourth inside tile below I:

   KLT.
   YSI.
   WAR.
   ....

However, since the N and D in GRAND are vertical to each other (4), there is no way for GRAND to be formed. Trying the second possible way to form SLY

   .L..
   Y.S.
   ....
   ....

given SWAY (2), with W an outside and A an inside letter, we would have

   .LW.
   YAS.
   ....
   ....

By clue 4, the outside letters N and D would be adjacent down the first column, with the F (FAITH) in the upper left corner:

   FLW.
   YAS.
   N...
   D...

Since outside letter T adjoins both inside letters I (FAITH) and S (SIGHTS), I would have to lie directly below S, with the fourth inside letter, R, to I's left:

   FLW.
   YAS.
   NRI.
   D...

The G in GRAND (4) would be the 2nd or 3rd letter across row 4. If G were the 2nd letter, then from SIGHTS H and T would fall

   FLW.
   YAS.
   NRIT
   DGH.

but there would be no way for ORBIT (1) to be made. If the G were the 3rd letter in row 4,

   FLW.
   YAS.
   NRI.
   D.G.

there again would be no way to form ORBIT. So, our third arrangement from above is the only one possible:

   ....
   ....
   ..SL
   ..Y.

From SWAY (2), since A is one of the four center letters, the A must be to the left of S; since W is an outer letter, it is directly under A:

   ....
   ....
   .ASL
   .WY.

From SIGHTS (2), T, an outside letter, is to the upper right of S. Given ASK in clue 2, K is in the lower-right corner; while from FAITH (1), I is directly above S. R, the fourth inside letter, is to I's immediate left:

   ....
   .RIT
   .ASL
   .WYK

From SIGHTS in clue 2, the GH must be the 2nd and 3rd or the 3rd and 4th letters across the top row of tiles. However, if G and H are the 2nd and 3rd letters, there isd no way for ORBIT (1) to be made. So, G and H are the 3rd and 4th tiles across row 1:

   ..GH
   .RIT
   .ASL
   .WYK

From ORBIT, B is to G's immediate left:

   .BGH
   .RIT
   .ASL
   .WYK

The N and D in GRAND (4) are the 2nd and 3rd or 3rd and 4th tiles down the first column. If they were 3rd and 4th, however, there would be no way to form THIRD (1). N is 2nd and D 3rd down the leftmost tile column:

   .BGH
   NRIT
   DASL
   .WYK

Finally, F is in the lower left corner (FAITH) and O in the upper left corner (ORBIT) (1). The Boggle board is as follows:

  OBGH
  NRIT
  DASL
  FWYK

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