1. Scientists have found a fragment of an ancient manuscript on mechanics. It was a piece of a book, the first page of which was numbered 435, and the last page was written with the same digits but in some other order. How many sheets were in this fragment?
Solution. Since the sheet has 2 pages and the first page is odd, the last page must be even. Therefore, the last digit is 4. The number of the last page is greater than the first. The only possibility left is 534. This means there are 100 pages in total, and 50 sheets.

Answer: 50.

Criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; 15 points - correct solution, but the number of pages (100) is recorded in the answer instead of the number of sheets; 10 points - the number of the last page is correctly determined, but the solution either stops there or is incorrect; 5 points - there are some correct ideas in the solution; **0** points - everything else.