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!).
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Puzzle 78
Hidden in the grid below are three boys names. Starting at any point, trace out the path of a chess knight. The last letter of one name starts the next one. What are the names?
Hint
As we don't know how of the people were library members, you can try a random number and see what happens.
Answer
4,200 books.
Reasoning
It doesn't matter how many library members there were, as 1/2 of them accepted the books, and each accepted 6 books, which is 3 books per member on average.
Similarly, it doesn't matter how many non-members there were as 3/4 of them accepted the books, and each accepted 4 books, which is again 3 books per non-member on average.
Double-Checking
Both the members and non-members accepted, on average, 3 books each.
And 1400 x 3 = 4200.
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Puzzle 80
Can you find the country hidden in the following sentence:
Alex's aches and pains felt worse because of swollen glands that had flared up during the night, perhaps a tonic would help?
