(related to Problem: Hannah's Puzzle)
Starting from any one of the $N$'s, there are $17$ different readings of $NAH,$ or $68$ ($4\times 17$) for the $4$ $N$'s. Therefore there are also $68$ ways of spelling $HAN.$ If we were allowed to use the same $N$ twice in a spelling, the answer would be $68\times 68,$ or $4,624$ ways. But the conditions were, "always passing from one letter to another." Therefore, for every one of the $17$ ways of spelling $HAN$ with a particular $N,$ there would be $51$ ways ($3\times 17$) of completing the $NAH,$ or $867$ ($17\times 51$) ways for the complete word. Hence, as there are four $N$'s to use in $HAN,$ the correct solution of the puzzle is $3,468$ ($4\times 867$) different ways.
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