Puzzle 2
If York is 9 miles away, London is 14 miles and Manchester is 23 miles away.
How far away is Leeds?
Puzzle Copyright © Kevin Stone
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Hint
Consonants and vowels (AEIOU) are each worth something.
Answer
12 miles away.
Consonants are worth 2 and vowels (AEIOU) are worth 3.
Double-Checking
York
3C + 1V = 3 x 2 + 3 = 9
London
4C + 2V = 4 x 2 + 2 x 3 = 14
Manchester
7C + 3V = 7 x 2 + 3 x 3 = 23
Leeds
3C + 2V = 3 x 2 + 2 x 3 = 12
Puzzle 3
As my autumnal birthday approaches I like to collect leaves! A little bizarre perhaps, but I enjoy it!
Starting on the first day of the month I collect 1 leaf, on the second day I collect 2 leaves, the third day I collect 3 leaves, and so on.
On my birthday, I will have collected 276 leaves altogether. Which day of the month is my birthday?
Bonus Question: how many days would it take for me to collect 56,616 leaves?
Puzzle Copyright © Kevin Stone
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Hint
How many leaves will I have collected on day 5?
Answer
On the 23rd.
Reasoning
We could simply keep adding until we get the required number:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23
= 276 leaves.
But a more mathematical method might help to answer the Bonus Question, as this might take a while if we keep adding!
So, let's create a method by imagining that we are adding the numbers from 1 to 30.
1 + 2 + 3 + … + 28 + 29 + 30
If we now take the numbers in pairs, taking one from each end, we have:
(1 + 30) + (2 + 29) + (3 + 28) + … + (15 + 16)
Each pair adds to 31, and we have 15 pairs. So the total sum is 31 x 15 = 465.
The total sum from 1 to any number (N) can be found using this technique, and we will have:
Each pair adds to (1 + N), and there are N ÷ 2 pairs. So the total is:
(1 + N) x N
—
2
In this puzzle, we know that this equals 276.
So:
(1 + N) x N = 276
—
2
We can expand the brackets, and multiply both sides by 2, to give:
N + N2 = 552
Rearranging we get:
N2 + N − 552 = 0
And 552 = 2 x 2 x 2 x 3 x 23, so this can be factorised as:
(N + 24) x (N − 23) = 0
Because we need to find a positive number of days, the only possible answer is:
(N − 23) = 0
So N = 23 days.
Bonus Question
To answer the bonus question, we have:
(1 + N) x N = 56616
—
2
Rearranging we get:
N2 + N − 113232 = 0
And 113232 = 24 x 3 x 7 x 337, so this can be factorised as:
(N − 336) x (N + 337) = 0
Because we need to find a positive number of days, the only possible answer is:
(N − 336) = 0
So N = 336 days (I did say that I liked collecting leaves!).
