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 18
At my local farmer's merchant, you can buy chicken feed for 15c per tonne, pig feed for $1.25 per tonne, and cattle feed for 40c per tonne. The feed can only be purchased by the tonne, and part tonnes aren't sold.
Last week I bought some animal feed, and luckily I managed to buy exactly 100 tonnes for exactly $100. How much of each feed did I buy?
Hint
What are the maximum and minimum tonnes of pig feed I could have bought?
Answer
I bought 15 tonnes of chicken feed, 75 tonnes of pig feed, and 10 tonnes of cattle feed.
15 x 0.15 = 2.25 75 x 1.25 = 93.75 10 x 0.40 = 4.00
100 tonnes = $100.00
A possible solution method follows …
I couldn't have bought 80 tonnes of pig feed because this would have cost the full $100.00, but I wouldn't have then had 100 tonnes in total.
Similarly, 70 tonnes of pig feed would have been too few, as this would cost $87.50 and even the remaining 30 tonnes all being cattle feed wouldn't have reached the required $100.00.
So, let's try 71 tonnes of pig feed = $88.75:
plus 1 x chicken feed + 28 x cattle feed is over $100.00 or 2 x chicken feed + 27 x cattle feed is under $100.00
so 71 tonnes of pig feed doesn't work.
Let's try 72 tonnes … and so on.
We soon get to 75 tonnes of pig feed and find the answer.
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Puzzle 19
Unluckily, last week I lost my measuring tape. It was unlucky as I had to measure Billy's toy box.
Luckily though I could remember that the top had an area of 720 square inches, the side was 800 square inches, and the end was 360 square inches.
