# Birthday Puzzle

• Mar 15th 2010, 11:04 PM
carriereg
Birthday Puzzle
Hello! some help on this puzzle please :)

For example there are 104 people in the room, state if the follow is true or false and

a) Explain the reasoning if true.
b) If false give a situation where the statement is not applicable.

1. There has to be at least two people in the room who have their birthday on the same day
I know there 365 days in a year

2. There has to be at least 9 people who have their birthdays in February

3. There has to be at least 10 people who have their birthdays in the same month

False

104 people/12 months = 8.6

4. There has to be at least 9 people who have their birthdays in the same month

False

104 people/12 months = 8.6

I'm a bit confused with part b) If false give a situation where the statement is not applicable.
So for the last two I could possibly say - if four more people joined the room then this would make the statement true?
• Mar 16th 2010, 06:16 AM
Focus
Quote:

Originally Posted by carriereg
Hello! some help on this puzzle please :)

For example there are 104 people in the room, state if the follow is true or false and

a) Explain the reasoning if true.
b) If false give a situation where the statement is not applicable.

1. There has to be at least two people in the room who have their birthday on the same day
I know there 365 days in a year

2. There has to be at least 9 people who have their birthdays in February

3. There has to be at least 10 people who have their birthdays in the same month

False

104 people/12 months = 8.6

4. There has to be at least 9 people who have their birthdays in the same month

False

104 people/12 months = 8.6

I'm a bit confused with part b) If false give a situation where the statement is not applicable.
So for the last two I could possibly say - if four more people joined the room then this would make the statement true?

About 4): Can 8.6 people have birthdays in one month? What is .6 of a person? Suppose you have 8 people having birthdays spread in 12 months (i.e. each month has 8 birthdays), that gives you 8 people left (104-12*8) over. How can you give them a month without making a month have 9 birthdays?

I think b) asks you to give a situation where it would be false. For example for 1) you can say, let them have birthdays in the first 104 days of the year, or for 2) you can collect together 104 people that were born in March. As with the above, you should be able to reason how you can select the people for number 3).
• Mar 16th 2010, 10:10 AM
Soroban
Hello, carriereg!

Quote:

There are 104 people in the room.
State if the following statements are true or false and

a) Explain the reasoning if true.
b) If false, give a situation where the statement is not applicable.
. . .
This means provide a counter-example, proving the statement false.
Quote:

1. There has to be at least two people in the room who have their birthday on the same day.
False.

The 104 people could have 104 different birthdays.
For example: from June 1 to September 12.

Quote:

2. There has to be at least 9 people who have their birthdays in February.
False.

They could all have their birthdays in August.

Quote:

3. There has to be at least 10 people who have their birthdays in the same month
False.

Their birth-months could be distributed like this:

. . $\displaystyle \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Jan} & \text{Feb} & \text{Mar} & \text{Apr} & \text{May} & \text{Jun} & \text{Jul} & \text{Aug} & \text{Sep} & \text{Oct} & \text{Nov} & \text{Dec} \\ \hline 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 5 \\ \hline \end{array}$

Quote:

4. There has to be at least 9 people who have their birthdays in the same month.
This is True.

We can try to contradict the statement
. . and place only 8 people in each month:

. . $\displaystyle \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Jan} & \text{Feb} & \text{Mar} & \text{Apr} & \text{May} & \text{Jun} & \text{Jul} & \text{Aug} & \text{Sep} & \text{Oct} & \text{Nov} & \text{Dec} \\ \hline 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\ \hline \end{array}$

But this accomodates only $\displaystyle 8 \times 12 \:=\:96$ people.

There are 8 more people who have birthdays in one of the twelve months.

Wherever they are placed, it makes a month with at least 9 people.