riddle of the day

Four runners, four statements and a single lie. Identifying which of them fails leads directly to the order of arrival.

Runners on a false trail

Reasoner
Master plays

Riddle statement

Four runners—Ana, Beto, Cora and Diego—came in positions from 1 to 4, without ties.

It is known that exactly one of these statements is false:

  • Ana: “I wasn't the first.”
  • Beto: “Ana came second.”
  • Cora: “Diego arrived immediately before me.”
  • Diego: “Beto came second.”

In what order did they arrive?

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Solution

Answer: they arrived in this order:

$ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{Beto},\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{Ana},\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{Diego},\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{Cora}. $

We know that exactly one statement is false.

Beto's sentence says that Ana came second. Let's assume it's true. So Ana occupies position 2.

Cora's sentence says that Diego arrived immediately before her. If Ana is already in position 2, the only possible block for Diego and Cora is: Diego 3rd, Cora 4th.

The remaining position, the first, is for Beto. Thus we obtain:

  1. Beto
  2. Anna
  3. Diego
  4. Cora

Now we verify the four statements:

  • Ana said: “I was not the first.” True.

  • Beto said: “Ana came second.” True.

  • Cora said: “Diego arrived immediately before me.” True.

  • Diego said: “Beto came second.” False.

There is exactly one false statement, as the statement requires. Therefore, the order is unique: Beto, Ana, Diego, Cora.