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 72
During a recent expedition, three intrepid adventurers were left stranded in the middle of the desert with only a crate full of apples.
During the night, Alex woke up and decided to hide some of the apples and hid a third, then promptly fell asleep again.
Billie woke up shortly after and also decided to hide a third of the remaining apples and then also dozed back to sleep.
Finally, Charlie woke up and seeing the others were asleep, took a third of what was left.
Of course, none of the adventurers knew of the other's antics, so, in the morning, they shared the remaining apples, each receiving sixteen. How many apples were in the crate originally?
Hint
How many apples were left after Charlie had taken some?
Answer
162 apples.
Alex hid 54, leaving 108.
Billie then hid 36, leaving 72.
Charlie then hid 24, leaving 48 available to share in the morning.
Reasoning
To work these numbers out, we start at the end and work backwards.
At the end there were 16 apples each, therefore there were 48 apples in total.
Charlie removed one third, leaving 48, so there must have been 72 apples before.
Billie removed one third, leaving 72, so there must have been 108 apples before.
Alex removed one third, leaving 108, so there must have been 162 apples before.
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