### Brain Teasers

# Men with Mustaches

You have a group of eighty two men, some of the men are tall, the number of the medium height men is double that of the tall men and the number of the short men is triple that of the medium height men and one man is very tall.

You know that all tall men are bald but not all of the short men are bald, and that some of the medium height men are bald but the number of the medium height men with hair is double of that of the bald medium height men. You also know that none of the tall men does not have mustache while none of the short men with hair have mustache, but half of the bald medium height men have mustaches and nine of the medium height men with hair have mustaches too. If the number of the bald short men with mustache is less by three than the rest of other men that have mustaches and the percentage of total men with mustaches is 50% then the question is: does the very tall man have mustache or not (do not guess) and how can you prove that?

You know that all tall men are bald but not all of the short men are bald, and that some of the medium height men are bald but the number of the medium height men with hair is double of that of the bald medium height men. You also know that none of the tall men does not have mustache while none of the short men with hair have mustache, but half of the bald medium height men have mustaches and nine of the medium height men with hair have mustaches too. If the number of the bald short men with mustache is less by three than the rest of other men that have mustaches and the percentage of total men with mustaches is 50% then the question is: does the very tall man have mustache or not (do not guess) and how can you prove that?

### Answer

Yes, the very tall man has a mustache.Let number of tall men = X then number of medium height men is 2X and number of short men is 6X (3*2X).

Total number of men = X+2X+6X+1(very tall man) = 82

Solving 9X = 81 gives:

Number of tall men is 9

Number of medium height men is 18

Number of short men is 54

And 1 very tall man

Grouping facts:

1. All tall men are bald and none of the tall men does not have mustache (in other words all of them have mustaches) that means you have 9 tall bald people with mustache.

2. The number of medium height men with hair is double of the number of bald medium height men (assume it Y) gives: Y+2Y = 18, and so Y = 6 which is the number of bald medium height men and the number of medium height men with hair = 12.

9 medium height men with hair have mustaches and 3 bald medium height men (half of them) have mustaches too.

3. Adding up the number of men that have mustaches up till this point from 1 & 2 above:

Number of men with mustaches = 9(tall) + 9(medium height men with hair) + 3(bald medium height men = 21

If the very tall man does not have a mustache, then the number of bald short men mustache = 18 (less by 3 than the above number.) If the very tall man does have a mustache then the number of bald short men is 19 (less by 3 than the above number plus one for the very tall man.)

Percentage of total number of men with mustaches = 0.5

= 0.5 * 82 = 41 men

There must be 21 + (21 + W) - 3 + W = 41 men with mustaches (where W is the one very tall man if having a mustache, and either equals 1 if he has a mustache or = 0 if he does not have one.)

Solving the equation: 42 + 2W - 3 = 41 gives that W = 1 and thus the very tall man has a mustache.

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