One thief entered a cave where he found a pile of diamonds. At first he…
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One thief entered a cave where he found a pile of diamonds. At first he thought of taking all of them, but listening to his conscience, he decided to take only half of them. While returning, he picked up two more diamonds from the pile. After that, a second thief entered the cave, took half of the remaining diamonds, and also picked up two more while returning. A third thief entered and did the same, and so did a fourth thief. When the fifth thief entered, he found only one diamond left in the cave. How many diamonds were there in the cave at first?
- A.
76
- B.
67
- C.
87
- D.
43
Attempted by 7 students.
Show answer & explanation
Correct answer: A
Concept: When a quantity is built up through a repeated forward operation (take half, then add or remove a fixed amount) and only the final result is known, the starting value is recovered by inverting the operation and applying it once per step in reverse order. Here, one thief's turn takes an amount "x" down to a remainder of x/2 − 2 (half taken, then two more removed while leaving); inverting this gives x = (remainder + 2) × 2 — applied once per thief, walking backward from a later known count to an earlier one.
Step-by-step (working backward from the fifth thief's find):
The fifth thief found 1 diamond — this is the amount left after the fourth thief's turn.
Reverse the fourth thief's turn: earlier amount = (1 + 2) × 2 = 6.
Reverse the third thief's turn: earlier amount = (6 + 2) × 2 = 16.
Reverse the second thief's turn: earlier amount = (16 + 2) × 2 = 36.
Reverse the first thief's turn: earlier amount = (36 + 2) × 2 = 76.
Cross-check (simulate forward from 76):
Start from 76: the first thief takes half (38) and picks up 2 more while leaving, removing 40 in total, leaving 76 − 40 = 36.
The second thief takes half of 36 (18), plus 2 more, removing 20 in total, leaving 36 − 20 = 16.
The third thief takes half of 16 (8), plus 2 more, removing 10 in total, leaving 16 − 10 = 6.
The fourth thief takes half of 6 (3), plus 2 more, removing 5 in total, leaving 6 − 5 = 1 — exactly matching what the fifth thief found, confirming the answer.
Answer: The cave originally held 76 diamonds.