Let us confine our attention to the L in the top left-hand corner.  Suppose we go by way of the E on the right:  we must then go straight on to the V, from which letter the word may be completed in four ways, for there are four E’s available through which we may reach an L. There are therefore four ways of reading through the right-hand E. It is also clear that there must be the same number of ways through the E that is immediately below our starting point.  That makes eight.  If, however, we take the third route through the E on the diagonal, we then have the option of any one of the three V’s, by means of each of which we may complete the word in four ways.  We can therefore spell LEVEL in twelve ways through the diagonal E. Twelve added to eight gives twenty readings, all emanating from the L in the top left-hand corner; and as the four corners are equal, the answer must be four times twenty, or eighty different ways.

256.—­THE DIAMOND PUZZLE.

There are 252 different ways.  The general formula is that, for words of n letters (not palindromes, as in the case of the next puzzle), when grouped in this manner, there are always 2^(n+1) — 4 different readings.  This does not allow diagonal readings, such as you would get if you used instead such a word as DIGGING, where it would be possible to pass from one G to another G by a diagonal step.

257.—­THE DEIFIED PUZZLE.

The correct answer is 1,992 different ways.  Every F is either a corner F or a side F—­standing next to a corner in its own square of F’s.  Now, FIED may be read from a corner F in 16 ways; therefore DEIF may be read into a corner F also in 16 ways; hence DEIFIED may be read through a corner F in 16 x 16 = 256 ways.  Consequently, the four corner F’s give 4 x 256 = 1,024 ways.  Then FIED may be read from a side F in 11 ways, and DEIFIED therefore in 121 ways.  But there are eight side F’s; consequently these give together 8 x 121 = 968 ways.  Add 968 to 1,024 and we get the answer, 1,992.

In this form the solution will depend on whether the number of letters in the palindrome be odd or even.  For example, if you apply the word NUN in precisely the same manner, you will get 64 different readings; but if you use the word NOON, you will only get 56, because you cannot use the same letter twice in immediate succession (since you must “always pass from one letter to another”) or diagonal readings, and every reading must involve the use of the central N.

The reader may like to find for himself the general formula in this case, which is complex and difficult.  I will merely add that for such a case as MADAM, dealt with in the same way as DEIFIED, the number of readings is 400.

258.—­ THE VOTERS’ PUZZLE.

THE number of readings here is 63,504, as in the case of “WAS IT A RAT I SAW” (No. 30, Canterbury Puzzles).  The general formula is that for palindromic sentences containing 2n + 1 letters there are (4(2^n -1)) squared readings.