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Ten Heaps, One Weighing (illustration)
FILE No.82Unsolved

Ten Heaps, One Weighing

Difficulty ★★★about 8 minTheft CaseWeighingNumbering
Ten heaps of coins, one all-fake. Fake coins weigh 9 g, real ones 10 g. You may weigh once.

Case Record

Confiscated coins are in ten heaps, each with plenty of coins. Exactly one heap is made entirely of fakes.

A genuine coin weighs 10 g and a fake 9 g. They look identical. You have one accurate digital scale and may weigh only once.

Your question

How can one weighing tell you which heap is fake?

Case Files

Evidence

Diagram

As many coins as the heap numberHeapCoins takenHeap 11Heap 22Heap 33......Heap 101055 coins in all. All genuine would be 550 g.
Diagram

Timeline

  1. Coins arrive in ten heaps.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

If you take the same number of coins from every heap, does the shortfall tell you which heap?

Open hint 2

What if you take a different number from each heap?

Open hint 3

Match the shortfall in grams to the heap number.

Solve It

Your question

How can one weighing tell you which heap is fake?

Choose the answer closest to the truth.

Share: I'm solving this

The Truth

Take 1 coin from heap 1, 2 from heap 2, ... 10 from heap 10, weigh them all together, and read the heap from how many grams light

The answer is: take 1 coin from heap 1, 2 from heap 2, ... 10 from heap 10, weigh them all together, and read the heap from how many grams light.

You take 1 + 2 + ... + 10 = 55 coins. If all were genuine they would weigh 550 g. If heap k is fake, k fake coins are in the pile, so the total is k grams light. A weight of 547 g means heap 3 is fake.

Grams light = heap numberWeightFake heap549 gHeap 1547 gHeap 3545 gHeap 5540 gHeap 10
Diagram of the solution

Key Points

  • Take a different number of coins from each heap
  • The shortfall equals the heap number
  • Pack the information into one total

Why people get fooled

Taking the same number from each heap makes it impossible to tell the heaps apart.

Similar tricks

  • The numbers-into-weight type
  • The one-total type

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