Puzzle 21
Can you find a five-digit number that has no zeros, and no digit is repeated, where:
The first digit is a prime number.
The second digit is the fifth digit minus the first digit.
The third digit is twice the first digit.
The fourth digit is the third digit plus three.
The fifth digit is the difference between the first digit and the fourth digit.
Don't forget that 1 isn't prime, but 2, 3, 5 and 7 are prime.
Puzzle Copyright © Kevin Stone
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Hint
Don't forget that 1 isn't prime.
Answer
23,475.
13,254 isn't allowed because 1 isn't prime.
Puzzle 22
My local bus company has recently expanded and no longer has enough room inside its garage for all of its buses.
Twelve of their buses have to be stored outside.
If they decide to increase their garage space by 40% they will have enough room for all of their current buses, plus enough room to store another twelve in the future.
How many buses does the company currently own?
Puzzle Copyright © Kevin Stone
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Hint
Think about the 40% and how many buses that represents.
Answer
72 buses.
Reasoning
If they increase the garage space by 40% this will give them enough room to hold the 12 buses that are currently outside, plus 12 more buses in the future.
Therefore, the 40% increase is enough space for 24 buses.
This means that 40% of the current garage capacity is 24 buses.
So 10% of the current garage capacity is 6 buses.
And 100% of the current garage capacity is 60 buses.
If we add the 12 buses stored outside we have a total of 72 buses.
Double-Checking
They currently have 72 buses, 12 of which are outside, so 60 are inside. Increasing their garage space (60 buses) by 40% will give room for 84 buses. Which is the 72 they currently have, with spaces for 12 more buses in the future. This confirms the answer.
Puzzle 23
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.
In total, 97 students study chemistry and physics, 138 students study physics and biology, and 152 students study chemistry and biology.
73 students study all three disciplines.
Can you determine how many students there are in the science department? The answer is not 814.
Puzzle Copyright © Kevin Stone
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Hint
How many students study chemistry and physics, but not biology?
Answer
500 students.
Reasoning
The answer is more easily seen if three intersecting circles are drawn, and the numbers inside each section are worked out.
We know that 73 students study all three disciplines, which allows us to work out the numbers.
97 students study chemistry and physics, so 97 - 73 = 24 study just chemistry and physics.
138 students study physics and biology, so 138 - 73 = 65 study just physics and biology.
152 students study chemistry and biology, so 152 - 73 = 79 study just chemistry and biology.
280 students study chemistry, so 280 - 73 - 24 - 79 = 104 study just chemistry.
254 students study physics, so 254 - 73 - 24 - 65 = 92 study just physics.
280 students study biology, 280 - 73 - 65 - 79 = 63 study just biology.
For a total of: 104 + 92 + 63 + 24 + 65 + 79 + 73 = 500 students.
Puzzle 24
In the following sum, the digits 0 to 9 have all been used.
O = Odd, E = Even (zero is even).
The top row's digits add to 9.
Can you determine each digit?
Puzzle Copyright © Kevin Stone
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Hint
The first digit in each row is even, so the first column can only be 2 + 8, 4 + 6, 4 + 8, 6 + 8 (either way around, and it must be a total over 9).
Answer
423 + 675 = 1098.
Reasoning
Remembering that:
even + even = even
odd + odd = even
even + odd = odd
To discuss individual letters, it's easiest to represent the sum as:
A B C
D E F +
————————
G H I J
A and D are both even, and can only be 2 + 8, 4 + 6, 4 + 8, 6 + 8 (either way around, and it must be a total over 9), so the carry can only be 1, so G = 1.
Since column 2 is even + odd = odd, there can be no carry from column 1 (since even + odd + 1 is always even). Therefore, neither C nor F (both odd) can be 7 or 9 (otherwise there would be a carry), so C and F are 3 and 5 (but we don't yet know which is which), therefore J = 8.
Because A + D is even there can't be a carry from column 2, therefore E can't be 9 as this would force a carry. Therefore, I = 9. So, B can't be 0. Therefore, H = 0.
The last remaining odd number makes E = 7. Making B = 2.
Therefore, A and D are 4 and 6 (but we don't yet know which is which).
Since the top row's digits have to add to 9 the top number must be 423.
This makes the sum 423 + 675 = 1098.
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