Puzzle 1
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: if I keep collecting one more leaf each day, 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!).
Puzzle 2
The following words have had their vowels (AEIOU) removed, can you find the missing animals?
rdvrk
rndr
prcpn
rhncrs
sqrrl
clt
frrt
sknk
dnky
ntlp
Puzzle Copyright © Kevin Stone
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Hint
One of the words is antelope.
Answers
rdvrk = aardvark
rndr = reindeer
prcpn = porcupine
rhncrs = rhinoceros
sqrrl = squirrel
clt = ocelot or colt
frrt = ferret
sknk = skunk or skink
dnky = donkey
ntlp = antelope
Puzzle 3
My friend's son, Billy, has the same number of brothers as sisters.
His sister, Laura, has twice as many brothers as she has sisters.
How many boys and girls are in the family?
Puzzle Copyright © Kevin Stone
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Hint
A little bit of algebra might help.
Answer
There are four boys and three girls.
If we call the number of boys B, and girls G.
From the first statement (we take one off as Billy is a boy):
B − 1 = G
and similarly:
2 x (G − 1) = B
Substitute G from the first equation into the second equation to give:
2 x ((B − 1) − 1) = B
2 x (B − 1) − 2 = B
2 x B − 2 − 2 = B
2 x B − 4 = B
B = 4
which means that B = 4 and hence G = 3.
Puzzle 4
How many squares of any size on this chessboard do not contain a rook?
Puzzle Copyright © Kevin Stone
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Hint
There are no larger squares than size 4 x 4.
Answer
There are 110 squares without a rook.
Reasoning
There are 60 squares of size 1 x 1.
There are 35 squares of size 2 x 2.
There are 12 squares of size 3 x 3.
There are 3 squares of size 4 x 4.
Giving a total of 60 + 35 + 12 + 3 = 110 squares.
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