Q: How many squares are on an 8 x 8 checkerboard? 10 x 10? How many rectangles?
8 x 8 There are:
64 1 x 1
49 2 x 2
36 3 x 3
25 4 x 4
16 5 x 5
9 6 x 6
4 7 x 7
1 8 x 8 for a total of 204 squares
For 10 x 10
100 1 x 1
81 2 x 2
64 3 x 3
49 4 x 4
36 5 x 5
25 6 x 6
16 7 x 7
9 8 x 8
4 9 x 9
1 10 x 10
for a total of 385 squares
The rectangle counting is significantly more challenging.
Click here for a beautiful explanation from Nigel Coldwell
Q: You have 8 batteries but only 4 of them work and 2 are needed to power a flashlight. What is the fewest number of pairs you need to test to guarantee you can get the flashlight on?
A: 7. If you break the batteries into 3 groups: Two groups of 3 and one group of 2. By doing this you guarantee that one of the groups has 2 working batteries. Both of the groups of 3 have 3 possible combinations of 2 batteries and the group of 2 only has 1 combination. So, 3 + 3 + 1 = 7 tries at most to find two working batteries.
Q: Can you take the clock face below, and cut it into four pieces such that the numerals on each part add up to the same number?
A: The key is to turn the number XI upside down to a IX or:
turn the IX upside down to a XI
The numbers 1-12 add up to 78 which is not divisible by 4
Replace 11 with a 9 and the numbers now add up to 76 which is divisible by 4
and/or replace the 9 with an 11 and now the numbers add up to 80 which is also divisible by 4
Q: Here is the Math/logic brain teaser that took the Internet by storm in mid-April 2015:
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates: May 15 May 16 May 19 June 17 June 18 July 14 July 16 August 14 August 15 August 17
Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively. Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.
So when is Cheryl’s birthday?
A: July 16th.
Can’t be a unique day. Therefore it cannot be May or June.
July 14 July 16
August 14 August 15 August 17
Has to be unique
August 15 August 17
If Bernard know and Albert now knows that it has to be July 16th. If it was August then Albert would not know.
(Thanks to Adam Schwartz for this solution!)
Q: A group of 100 students play various instruments: 70 play trombone, 75 play accordion, 85 play violin and 80 play guitar. What is the minimum number of students who must play all 4?
A: 10 students is the minimum number that must play all 4 instruments
Here we need to look for the overlap since there must be some students who play more than one instrument. The best way to do this is with a Venn diagram (which was really popular during the New Math of the ’70’s when I was in school!).
The Math below is done to find the minimum number of students who must play all 4:
So if 70 play trombone and 75 play accordion (which add up to 145) there must be at least a 45 overlap that play both.
We will now consider those 45 (that play 2 instruments) with the 85 who play violin (which add up to 130) so there must be a 30 overlap that play 3.
We will now consider those 30 with the 80 who play guitar (which add up to 110) so there must be a 10 overlap that play all 4 instruments.