<p>In Ancient Times, Four Mathematicians Discovered a Game of L</p>

20.

In Ancient Times, Four Mathematicians Discovered a Game of L

In ancient times, four mathematicians discovered a game of logic. The four people were Mr. Macdonald, Mr. Franke, Mr. Kennedy, and Mrs. Sharma. Mr. Kennedy seated the other three people one behind the other so that Mr. Macdonald saw Franke and Sharma and so that Franke and Sharma saw no one.
Mr. Kennedy had 5 hats which he showed to the mathematicians. Three of the hats were grey and 2 white. Mr. Kennedy put a hat on each person's head and hid the remaining two hats in the box. "What colour is your hat, Mr. Macdonald?" asked Mr. Kennedy. Mr. Macdonald did not know. Mr. Franke did not know the colour of his hat either when asked.
Mrs. Sharma, who couldn't see any hats at all, gave the correct answer when asked the colour of her hat. What was the colour of her hat and how did she figure it out?

Submitted by tartle · Added 17 January 2013 · Updated 5 July 2026

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Solution:

Mrs. Sharma's hat must be grey. Mr. MacDonald could not determine his hat color, which means he saw either one grey and one white hat or two grey hats on Mr. Franke and Mrs. Sharma. Mr. Franke, seeing Mrs. Sharma, could not determine his hat color either, indicating that Mrs. Sharma must be wearing a grey hat. Since Mr. Franke could not confidently identify his hat color, Mrs. Sharma deduced that her hat must be grey.

Comments (4)

leanard0 13 April 2013

The only way Mr. Macdonald could not have known his hat color is if he saw more than one black hat in front of him. Mr. Franke realizing this would only be uncertain of his own hat color if the hat in front of him was black. Hence Mrs. Sharma could confidently say her hat was black

arun229 29 April 2014

mr.leanard0.....there is not at all black colour in the hats mentioned..... :shock: :)

rgsthegreat 16 March 2015

The only way MacDonald could've answered is if he saw two grey hats. Now that he didn't answer, he either saw 'two white hats' or 'one white and one grey hat'

Now it would've been easy for Franke to answer it if he saw a grey hat on Sharma's head coz the possibility is WW and GW.

Seeing Franke not answer, Sharma could easily derive that she was wearing a white hat. :D

king29 ★ Solved 22 September 2015

Sharma is wearing a Grey hat.

MacDonald: He could see Frank and Sharma. So, he could have definitely answered if both of them are wearing white (then his hat would be grey). As he did not answer they could be wearing one grey and one white or both grey.

Now coming to Franke: He could see only Sharma. So, if Sharma is wearing white hat, he could have easily answered he is wearing grey based on MacDonald's cases. So he see's Sharma wearing Grey and he could not figure out his hat is White/Grey.

Sharma: As Franke could not answer, she is not wearing white hat. That means she is wearing Grey one.

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