There are four Answers coming; How do I do this?

Question

Q. Sam tore out some pages from a book. The number of the $1^{st}$ page he tore out was $183$ and it is known that the number of the last page he tore out is written with the same digits in some order. How many pages did Sam tear out?

My Attempt:
The number of possibilities for the last page was $3! = 6$
Now, page no. $138$ can be discarded since Sam cant go before $183$.
But the other 4 possibilities seem to give 4 different answers. How do I do it?
Do I have to consider all four possibilities or is there another approach to the answer?

If you tear page $183$ is the page no on front side or back side?? — , Dec 28 '13 at 11:05
There might even be more than four. The question doesn't say that the pages came out in one go as one chunk. He could have torn them out one-by-one at random. — Fly by Night, Dec 28 '13 at 11:15

score 3 · Accepted Answer · answered Dec 28 '13 at 11:14

3

Since the number of the first page he tore was 183, it is safe to assume that 183 was written on the front.

Now assume Sam tore an odd-numbered page as the last. Howevever, to tear that page he must also tear the page right behind it, making the n+1 page the page last torn. So he cannot have torn an odd numbered page last.

We know he has torn an even numbered page last which is either 138 or 318. You've already shown that it can't be 138 so the last page torn must be 318.

answered Dec 28 '13 at 11:14

ulucs

186

Cute. A nice exercise in nonlinear thinking. – Nate Eldredge Dec 28 '13 at 12:00
Thanks! That was a great help. – Anish Bhattacharya Dec 28 '13 at 17:34

There are four Answers coming; How do I do this?

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