Puzzle 37
I am compiling the new BrainBashers world almanac, and it now contains lots more pages.
I know that it takes 333 digits to print the page numbers in sequence.
How many numbered pages does the book have?
How many times does the number 3 appear?
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Hint
Think of the pages from 1-9, and then 10-99.
Answers
There are 147 pages.
The number 3 appears 35 times.
Answer #1
For pages 1-9, there are 9 pages, which is 9 digits.
For pages 10-99, there are 90 pages, which is 180 digits.
This is a total so far of 189, therefore we require another 333 – 189 = 144 digits, which is another 144 ÷ 3 = 48 pages.
Taking us to 9 + 90 + 48 = 147 pages in total.
Answer #2
From page 1 to 147 we have 15 pages that end in 3:
3, 13, 23, …, 143
We also have the ten pages that start with 30:
30, 31, 32, …, 39
Plus the ten pages that start with 130:
130, 131, 132, …, 139
For a total of 15 + 10 + 10 = 35 number 3's.
Puzzle 38
During a recent BrainBashers thinking contest(!), the total number of points scored by the first six players was 103 and every score was above zero.
The first player scored half the points of the second player, who in turn scored 6 points fewer than the third player.
The third player in turn scored two thirds the points of the fourth player.
The fifth player managed to score the same number of points as the difference between the first and fourth player's points.
Finally, the sixth player scored 14 fewer points than the fifth player.
Can you determine how many points the sixth player managed to score?
Puzzle Copyright © Kevin Stone
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Hint
The fourth player is the key to this tricky question.
Answer
9 points.
Respectively the scores were 7, 14, 20, 30, 23, 9.
Reasoning
If we label the six players A, B, C, D, E, and F, we know that:
[1] A + B + C + D + E + F = 103
and from the clues:
A = B ÷ 2
B = C – 6
C = D x 2 ÷ 3
E = D – A
F = E – 14
If instead we choose E = A – D, we'd later see that we end up with a negative value for E, which isn't allowed.
Since D is the letter we're missing information for, it's best to find all of the other letters in terms of D.
These steps are left as an exercise (use C in the equation for B, etc), but the result is:
A = ( D – 9) ÷ 3
B = (2D – 18) ÷ 3
C = (2D ) ÷ 3
D = (3D ) ÷ 3
E = (2D + 9) ÷ 3
F = (2D – 33) ÷ 3
Writing it as D = 3D ÷ 3 makes it slightly clearer to see, when adding in the next step.
We can then use these in [1] to find that 12D = 360, so D = 30.
Therefore, F = 9.
Puzzle 39
What four-digit number has …
… Digit 2 smaller than Digit 4 which is two thirds of Digit 1 which is two thirds of Digit 3 which is three times Digit 2?
Puzzle Copyright © Kevin Stone
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Hint
Look at Digit 3 first.
Answer
6394.
Reasoning
Labelling the clues:
Digit 2 is smaller than Digit 4
Digit 4 is two thirds of Digit 1
Digit 1 is two thirds of Digit 3
Digit 3 is three times Digit 2
By (3), Digit 3 is divisible by 3:
Digit 3 = 3, 6, 9
By (3), Digit 1 is two thirds of Digit 3:
Digit 1 = 2, 4, 6.
But, by (2), Digit 1 is divisible by 3:
Digit 1 = 6, and Digit 3 = 9.
By (4), Digit 3 is three times Digit 2:
Digit 2 = 3.
By (2), Digit 4 is two thirds of Digit 1:
Digit 4 = 4.
Puzzle 40
Can you find the number that best completes this puzzle?
Puzzle Copyright © Kevin Stone
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Hint
The answer involves a square root.
Answer
4.
The middle of each triangle contains the square root of the sum of the three surrounding numbers.
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