The Library Borrowing Puzzle
Four students - Ana, Ben, Cara and Dev - each borrowed a different number of books from the library this term. The totals were 3, 5, 8 and 12, in some order. Clues:
- Ana borrowed more than Ben but fewer than Cara.
- Dev borrowed an odd number of books.
- The student who borrowed 12 books is not Cara.
- Ben borrowed fewer than 5 books.
A librarian also notes that anyone who borrowed 8 or more books received a bookmark, and exactly two students received bookmarks.
1. Work out how many books each student borrowed. Explain each deduction.
Read the worked answer
From clue 4, Ben borrowed fewer than 5, so Ben = 3 (the only total under 5).
From clue 1, Ana is between Ben and Cara, so Ben < Ana < Cara.
Remaining totals for Ana, Cara, Dev are 5, 8, 12.
Dev is odd (clue 2), so Dev = 5.
That leaves 8 and 12 for Ana and Cara, with Ana < Cara, so Ana = 8 and Cara = 12.
Check clue 3: the 12-book borrower is Cara - this contradicts clue 3.
So the clues as written are inconsistent. Re-examining: if clue 3 holds (Cara ≠ 12), then Cara = 8 and Ana = 12, but that breaks Ana < Cara. The puzzle has no solution unless one clue is wrong. The bookmark note (exactly two students at 8+) is consistent with either arrangement, so it does not resolve the conflict.
2. Identify which single clue must be removed or altered to make the puzzle solvable, and give the resulting solution.
Read the worked answer
The conflict is between clue 1 (Ana < Cara) and clue 3 (Cara is not the 12-borrower), once Ben = 3 and Dev = 5 are fixed.
With Ana and Cara holding 8 and 12:
- If Ana < Cara, then Cara = 12, which violates clue 3.
- If clue 3 holds, Cara = 8 and Ana = 12, which violates clue 1.
So either clue 1 or clue 3 must change.
Option A - remove clue 3: Solution is Ben 3, Dev 5, Ana 8, Cara 12. Bookmark holders (8+): Ana and Cara - exactly two, consistent.
Option B - remove clue 1: Solution is Ben 3, Dev 5, Cara 8, Ana 12. Bookmark holders: Cara and Ana - still exactly two, consistent.
Both repairs are valid; the bookmark note cannot choose between them, so the puzzle is under-determined even after a fix. A good answer states this rather than guessing.