Dr. Johnson decided to check if his physician assistant Anna is smart enough. He prepared three envelopes and put a cheque for $100 in one of the three envelopes. On the first envelope he wrote “This envelope does not have the cheque”, on the second envelope he wrote “This envelope has the cheque”, and on the third one he wrote “The second envelope does not have the cheque”.  Then he told Anna that if she wants to get the cheque, she needs to guess which envelope contains the cheque, without touching the envelopes, just by reading the writings on the envelopes. He also said that only one writing was true and two others were false.  Which envelope does Anna need to pick up to get the cheque?

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Answer:

Anna needs to pick up the first envelope.

If the statement on the second envelope is true (which means that the second envelope contains the cheque), then the statements on the first and third envelopes are false. If the statement on the first envelope is false then the first envelope contains thecheque , but we have assumed that the second envelope contains the cheque, so we get a contradiction. Therefore, the statement on the second envelope must be false which means that the second envelope does not have the cheque. Hence, the statement on the third envelope is true, and, therefore, the statement on the first envelope is false. This means that the first envelope has the cheque.

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