Maths Puzzles

Puzzle 1

Within the BrainBashers school, the science department has three disciplines.
In total, 280 students study chemistry, 254 students study physics and 280 students study biology.
97 students study both chemistry and physics, 138 students study both physics and biology, 152 students study both chemistry and biology.
73 students study all three disciplines.
Can you determine how many students there are in the science department? The answer is well below 814.

[Ref: ZZYU] © Kevin Stone

Hint: How many students study chemistry and physics, but not biology?

Answer: 500 students.

The answer is most easily seen if three intersecting circles are drawn, and the numbers inside each section worked out.

Puzzle 2

What value of * makes the following correct:

[Ref: ZWPF] © Kevin Stone

Hint: The answer is a whole number less than 10.

Answer: 3.

Checking that 3 works:

3 3 3 3
- - - + - = -
3 2 4 12
1 - 1.5 + 0.75 = 0.25
0.25 = 0.25, which confirms that 3 is correct.
To deduce the answer directly from the question, we can multiply throughout by 12,
* * * *
- - - + - = -
* 2 4 12
to give:
12 - 6* + 3* = 1*
12 - 3* = 1*
12 = 4*
3 = *
As required.

Puzzle 3

Starting in the bottom left corner and moving either up or right, one square at a time, adding up the numbers along the way, what is the largest sum which can be made once you have reached the top right corner?

[Ref: ZNGR] © Kevin Stone

Hint: The answer is over 35.

Answer: 37.

Puzzle 4

What 5 digit number am I thinking of.

It has two prime digits.
Digit 3 is the highest digit.
Digit 2 is lowest digit.
Digit 1 is higher than the sum of digits 4 and 5.
Digit 5 is half of digit 4.
Digit 1 is one smaller than digit 3.
Digit 5 is between digit 2 and digit 1
There are no duplicates and the digit 0 doesn't appear.
Don't forget that 1 isn't prime.

[Ref: ZVHE] © Kevin Stone

Hint: What can digits 4 and 5 be?

Answer: 71842.
The number can be represented as:
_ _ _ _ _
Since digit 5 is half of digit 4 we have four possibilities:
_ _ _ 2 1
_ _ _ 4 2
_ _ _ 6 3
_ _ _ 8 4
But since digit 1 is higher than the sum of digit 4 and digit 5 we can eliminate two possibilities leaving:
_ _ _ 2 1
_ _ _ 4 2
Since digit 5 is between digit 2 and digit 1, we can eliminate _ _ _ 2 1, leaving:
_ _ _ 4 2
Since digit 1 must be higher than the sum of digit 4 and digit 5 we have three possibilities:
7 _ _ 4 2
8 _ _ 4 2
9 _ _ 4 2
But digit 1 is 1 smaller than digit 3 we can eliminate 9 _ _ 4 2, leaving
7 _ 8 4 2
8 _ 9 4 2
Since digit 2 is the lowest number, it must be 1:
7 1 8 4 2
8 1 9 4 2
The answer has exactly 2 prime digits, which only leaves
7 1 8 4 2. QED.

Puzzle 5

Daft Dave, our local DIY expert, is renovating the local hospital.
He has to number each of the rooms in the hospital with plastic numbers.
However, he has run out of the number 9.
How many 9's must Dave buy in order to number all 1,000 rooms?

[Ref: ZRQK] © Kevin Stone

Hint: Don't forget the extra 9's in 90, 91, etc.

Answer: 300.
Start with: 9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99 - which is 20 9's.
We then repeat this for 100, 200, 300, up to 900 - which is 200 9's.
But from 900 to 999 we have an extra 100 9's as each starts with a 9.
So the total is 300.