Puzzle 21
Which circle, square, and triangle, have the closest area to the blue doughnut shape?
The drawings are to scale, so you might be able to judge by eye, or you can work out the actual areas.
Puzzle Copyright © Kevin Stone
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Hint
The area of a circle is π x Radius2.
Answers
Circle: 35.
Square: 31.
Triangle: 47.
Doughnut
The area of a circle is π x Radius2.
The larger circle has diameter = 40, therefore the radius is 20, and the area is π x 202 = 400π.
The smaller circle has diameter = 20, therefore the radius is 10, and the area is π x 102 = 100π.
Therefore the shaded area is 400π – 100π = 300π ≈ 942.
Circle
The area of a circle is π x Radius2.
The circle with diameter 31 has a radius of 15.5 and an area of π x 15.52 ≈ 755.
The circle with diameter 35 has a radius of 17.5 and an area of π x 17.52 ≈ 962 (closest match).
The circle with diameter 39 has a radius of 19.5 and an area of π x 19.52 ≈ 1195.
Square
The area of a square is Side x Side.
The square with side 31 has an area of 31 x 31 = 961 (closest match).
The square with side 35 has an area of 35 x 35 = 1225.
The square with side 39 has an area of 39 x 39 = 1521.
Triangle
The area of a triangle is 1/2 x Base x Height. Using Pythagoras' theorem it can be shown that the area of an equilateral triangle is √3 x Base2 ÷ 4.
The triangle with side 39 has an area of √3 x 39 x 39 ÷ 4 ≈ 659.
The triangle with side 43 has an area of √3 x 43 x 43 ÷ 4 ≈ 801.
The triangle with side 47 has an area of √3 x 47 x 47 ÷ 4 ≈ 957 (closest match).
Puzzle 22
Start in the bottom left corner and move either up or right, one square at a time, adding up the numbers. What is the largest total you can make?
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Note: this puzzle is not interactive, and the numbers cannot be clicked.
Puzzle Copyright © Kevin Stone
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Hint
The answer is over 35.
Answer
The largest total that can be made is 37.
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Puzzle 23
At the local sweet shop, three particularly nice sweets are on special offer.
A Nobbler is over three times the price of a Sparkle.
Six Sparkles are worth more than a Wibbler.
A Nobbler, plus two Sparkles costs less than a Wibbler.
A Sparkle, a Wibbler and a Nobbler together cost 40p.
Can you determine the price of each type of sweet?
Puzzle Copyright © Kevin Stone
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Hint
Which sweet is the least expensive?
Answer
Sparkle = 4p
Wibbler = 23p
Nobbler = 13p
Reasoning
By (3) a Nobbler, plus two Sparkles costs less than a Wibbler, therefore a Wibbler must be the most expensive sweet.
By (1) a Nobbler is over three times the price of a Sparkle, therefore a Sparkle must be the cheapest sweet.
So the order of sweets, from the least to most expensive, is Sparkle, Nobbler, Wibbler.
If a Sparkle was 1p, by (2) a Wibbler could only be up to 5p, by (4) a Nobbler would cost at least 34p, which is more than a Wibbler and isn't allowed as the Wibbler is the most expensive sweet.
If a Sparkle was 2p, by (2) a Wibbler could only be up to 11p, by (4) a Nobbler would cost at least 27p, which is more than a Wibbler and isn't allowed as the Wibbler is the most expensive sweet.
If a Sparkle was 3p, by (2) a Wibbler could only be up to 17p, by (4) a Nobbler would cost at least 20p, which is more than a Wibbler and isn't allowed as the Wibbler is the most expensive sweet.
So a Sparkle must be at least 4p.
If a Sparkle was 4p, by (1) a Nobbler must be at least 13p, by (4) a Wibbler would cost 23p – this combination matches all of the clues and is a possible solution.
If a Sparkle was 4p and a Nobbler 14p, by (4) a Wibbler would cost 22p. This would not satisfy (3). And if we increase the price of a Nobbler, (3) is never satisfied.
If a Sparkle was 5p, by (1) a Nobbler must be at least 16p, by (4) making a Wibbler at most 19p. This would not satisfy (3).
If we increase the price of a Sparkle or Nobbler further, (3) is will never be satisfied.
Therefore, the only solution we came across must be the correct one.
Puzzle 24
A coin collector decides to divide his coin collection between his children.
The eldest gets 1/2 of the collection, the next gets 1/4 of the collection, the next gets 1/5 of the collection, and the youngest gets the remaining 49 coins.
How many coins are in the collection?
Puzzle Copyright © Kevin Stone
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Hint
What do the fractions add up to?
Answer
There are 980 coins in the collection.
Reasoning
Using fractions:
1/2 + 1/4 + 1/5 = 19/20
The remaining 1/20 is 49 coins.
Therefore, the 20/20 must equal 20 lots of 49 = 980.
Alternative Reasoning
Using percentages:
50% + 25% + 20% = 95%
The remaining 5% is 49 coins.
If 5% is 49 coins, 10% is 98 coins, 100% is 980 coins.
Double-Checking
1/2 of 980 is 490
1/4 of 980 is 245
1/5 of 980 is 196
and the remaining 49 coins.
And 490 + 245 + 196 + 49 = 980.
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