Puzzle 293
Summer Paintings - Logic Puzzles
At a recent painting competition, the results were mislaid. Luckily, I remembered the following:
Emery's rendition of the Constable painting was not in last place.
The person who painted the Monet was very successful and took first place.
Glen only just managed to avoid last place and came third.
Alex beat the person who painted the Taylor, and the person who painted the Van Gogh beat Drew.
Can you determine who painted what, and who won?
Puzzle Copyright © Kevin Stone
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Hint
Which painting won, and who came first?
Answer
# Name Artist
1 Alex Monet
2 Emery Constable
3 Glen Van Gogh
4 Drew Taylor
Reasoning
The four entrants were: Alex, Drew, Emery, Glen.
The four artists were: Constable, Monet, Taylor, Van Gogh.
By (2), the Monet came first.
1 Monet
2
3
4
By (3), Glen came third.
1 Monet
2
3 Glen
4
By (1), the Constable can't have been last, so must have been second (as Emery painted it, and not Glen).
1 Monet
2 Constable
3 Glen
4
The Taylor was either third or fourth, either way, by (4) Alex must have won.
1 Alex Monet
2 Constable
3 Glen
4
By (4), the Van Gogh came third, leaving Drew in last place.
1 Alex Monet
2 Constable
3 Glen Van Gogh
4 Drew
Leaving Emery in second place, and the Taylor in last place.
1 Alex Monet
2 Emery Constable
3 Glen Van Gogh
4 Drew Taylor
Puzzle 294
I have a bottle and a straw which cost me 60 pence. The bottle cost 10 pence more than the straw. What does each item cost?
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Answer
The bottle cost 35 pence and the straw cost 25 pence.
Puzzle 295
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!).
Puzzle 296
Entry Exam Section B - Logic Puzzles
Here is a snippet of section B of the curious multiple-choice entrance exam into the exclusive BrainBashers puzzle club.
Q1. The answer to Q2 is:
A. B
B. A
C. D
D. C
Q2. The answer to Q3 is:
A. C
B. D
C. B
D. A
Q3. The answer to Q4 is:
A. D
B. A
C. C
D. B
Q4. The answer to Q1 is:
A. D
B. C
C. A
D. B
Puzzle Copyright © Kevin Stone
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Hint
Can Q1's answer be A?
Answer
1. D
2. C
3. B
4. A
The easiest way to solve this puzzle is to use Q1 to check the logic.
If Q1's answer was A, this tells us that Q2's answer is B, which tells us Q3's is D, which tells us Q4's answer is B, which tells us Q1's answer is C. A contradiction.
If Q1's answer was B, this tells us that Q2's answer is A, which tells us Q3's is C, which tells us Q4's answer is C, which tells us Q1's answer is A. A contradiction.
If Q1's answer was C, this tells us that Q2's answer is D, which tells us Q3's is A, which tells us Q4's answer is D, which tells us Q1's answer is B. A contradiction.
If Q1's answer was D, this tells us that Q2's answer is C, which tells us Q3's is B, which tells us Q4's answer is A, which tells us Q1's answer is D. Which is correct, and therefore the correct answer.
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