Puzzle 1
If I give you seven apples, you will then have five times as many as I would then have.
However, if you give me seven apples, we will then both have the same number of apples.
How many apples do I currently have?
Puzzle Copyright © Kevin Stone
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Hint
A little bit of algebra might help.
Answer
I have 14 apples and you have 28 apples.
Reasoning
We can work this out with a little algebra. I have A apples and you have B apples, then
[1] 5 x (A – 7) = B + 7
[2] A + 7 = B – 7
From [1] we have:
5A – 35 = B + 7
[3] 5A = B + 42
From [2] we have:
[4] A + 14 = B
We can then use [4] in [3] to give:
5A = (A + 14) + 42
5A = A + 56
4A = 56
A = 14
Which gives B = 28 (by [4]).
Puzzle 2
During a recent BrainBashers thinking contest(!), the total number of points scored by the first six players was 103 and every score was above zero.
The first player scored half the points of the second player, who in turn scored 6 points fewer than the third player.
The third player in turn scored two thirds the points of the fourth player.
The fifth player managed to score the same number of points as the difference between the first and fourth player's points.
Finally, the sixth player scored 14 fewer points than the fifth player.
Can you determine how many points the sixth player managed to score?
Puzzle Copyright © Kevin Stone
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Hint
The fourth player is the key to this tricky question.
Answer
9 points.
Respectively the scores were 7, 14, 20, 30, 23, 9.
Reasoning
If we label the six players A, B, C, D, E, and F, we know that:
[1] A + B + C + D + E + F = 103
and from the clues:
A = B ÷ 2
B = C – 6
C = D x 2 ÷ 3
E = D – A
F = E – 14
If instead we choose E = A – D, we'd later see that we end up with a negative value for E, which isn't allowed.
Since D is the letter we're missing information for, it's best to find all of the other letters in terms of D.
These steps are left as an exercise (use C in the equation for B, etc), but the result is:
A = ( D – 9) ÷ 3
B = (2D – 18) ÷ 3
C = (2D ) ÷ 3
D = (3D ) ÷ 3
E = (2D + 9) ÷ 3
F = (2D – 33) ÷ 3
Writing it as D = 3D ÷ 3 makes it slightly clearer to see, when adding in the next step.
We can then use these in [1] to find that 12D = 360, so D = 30.
Therefore, F = 9.
Puzzle 3
Every year, just before my birthday, I open my money box to see how much money I have saved.
Imagine my surprise when I found exactly £49.58 made up of equal numbers of 2p, 5p, 10p and 20p.
How many of each coin did I find?
Puzzle Copyright © Kevin Stone
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Hint
Remember that you have the same number of each coin.
Answer
134 of each coin.
Reasoning
We have equal numbers of 2p, 5p, 10p, and 20p.
And 2p + 5p + 10p + 20p = 37p.
There are 134 lots of 37p in £49.58.
Double-Checking
134 x 37 = 4958.
Puzzle 4
Can you find three consecutive prime numbers …
… that equal 409,457 when multiplied together?
Puzzle Copyright © Kevin Stone
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Hint
The cube root of the given number is a good starting point.
Answer
71, 73, 79.
Reasoning
The cube root of 409,457 is around 74.
The closest consecutive prime numbers to 74 are: 61, 67, 71, 73, 79, 83, 89, 97.
Using a calculator, it is quickly seen that the answer can only be found using 71, 73 and 79.
