THE DEIFIED PUZZLE.
(
Unicursal and Route Problems)
In how many different ways may the word DEIFIED be read in this
arrangement under the same conditions as in the last puzzle, with the
addition that you can use any letters twice in the same reading?
Answer:
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.
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