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Minimum Number of Heads with Same Number of Hairs Riddle

Minimum Number of Heads with Same Number of Hairs Riddle

In a city of known number of residents find the minimum number of heads with same number of hairs when maximum number of hairs is also known. Read on...

Minimum number of heads with Same number of hairs riddle

In a city of 7500000 residents someone may be totally bald, but some one else may have her head full of maximum number of hairs 500000. Can you find the minimum number of residents with same number of hairs on theirs heads?

Recommended time to solve: 15 minutes.

Solution to the minimum number of heads with same number of hairs puzzle

The puzzle is a bit confusing, but can we start solving by asking key questions and finding answers!

The first key question goes straight to the heart of the problem,

Question 1: On what does the minimum number of heads with same number of hairs depend?

Answer to the question takes a little careful thought to form. Nevertheless its sense is,

Answer 1: Minimum number of heads with same number hairs become less and less When number of heads WITH DIFFERENT NUMBER OF HAIRS BECOMES MORE AND MORE.

In other words more precisely,

Precise answer 1: the minimum number of heads with same number hairs is directly related to the maximum number of different count of hairs in the heads of the residents.

Now then we have to find out this maximum number of different hair counts.

Next question follows easily,

Question 2: What can be the different hair counts in the residents' heads? When does the maximum spread in this different hair counts happen?

Answer is also easy,

Answer 2: Maximum spread to the different hair counts happen only when the number of hairs TAKES ON EVERY VALUE FROM 0 TO THE MAXIMUM POSSIBLE 500000. This is total number of 500001 values of hair counts.

To aid visualization of the situation in our mind's eye, we equate this spread of 500001 number of hair counts to a series of boxes numbering 500001. It is shown below.

Pigeonhole boxes

Imagine these boxes as pigeonholes and the residents as pigeons occupying a particular hole say, 1002, when the all such residents have their hair count 1002. This way all 7500000 residents must have to be distributed among these 500001 pigeonholes. Isn't it?

In case all the pigeonholes have same number of pigeons, what does this number signify? It is just the quotient of dividing 7500000 by 500001.

Quotient of dividing 7500000 by 500000 is 15, but when 500000 is increased by 1, the quotient reduces to 14 giving the simple relation,

7,500,000 > 500,001 x 14 + 1

This means even if we place an equal number of 14 residents in each of the 500,001 pigeonholes, we will still be left with at least one resident without any pigeonhole.

Placing him in any of the pigeonholes (which are actually value of number of hairs on the heads of residents) makes the occupancy of that pigeonhole at least 15.

We have the answer finally. The minimum number of residents with same hair count is 15.

Fun isn't it!

End note

This in fact is a demonstration of the powerful mathematical principle known as Pigeonhole principle as well as solution of the puzzle by Question Analysis and Answer technique or QAA technique in brief.

This also is a good study on mathematical reasoning based on inequality analysis that in general is not natural to us.

Though this application of the simple but powerful Pigeonhole principle is mathematical, you never know when you may be able to solve a real life problem using this principle.

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