Puzzle 249
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?
Puzzle Copyright © Kevin Stone
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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 250
Mixed Hurdle Trials - Logic Puzzles
At the recent BrainBashers Mixed National Hurdle trials, ten contestants competed for the coveted BB Shield.
There were numerous judges who made sure that the results were accurate, from their notes can you determine who finished where?
Crystal Ball finished before May Eye and Mike Stup, while Leigh Vamessage was victorious over May Eye and Laura DeLand.
Jack Pott was beaten by Holly Day, Mike Stup, and Justin Time.
Laura DeLand was beaten by Mike Stup, May Eye, and Crystal Ball, whereas Joe King beat Holly Day and Jack Pott, but lost to Laura DeLand and Inga Hootz.
Leigh Vamessage was beaten by Crystal Ball and Inga Hootz.
Inga Hootz was beaten by Justin Time, who, in turn, lost to Crystal Ball, but beat Holly Day and Jack Pott.
May Eye finished ahead of Mike Stup and Holly Day (who lost to Leigh Vamessage and Inga Hootz).
Puzzle Copyright © Kevin Stone
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Hint
Rewrite the clues so that those who finished ahead are written first.
Answer
Crystal Ball (CB)
Justin Time (JT)
Inga Hootz (IH)
Leigh Vamessage (LV)
May Eye (ME)
Mike Stup (MS)
Laura DeLand (LD)
Joe King (JK)
Holly Day (HD)
Jack Pott (JP)
Reasoning
If we number the clues, write each name using two letters, and rewrite the clues so that those who finished ahead are written first:
CB beat ME and MS
LV beat ME and LD
HD, MS, and JT, all beat JP
MS, ME, and CB, all beat LD
JK beat HD and JP
LD and IH both beat JK
CB and IH both beat LV
JT beat IH
CB beat JT
JT beat HD and JP
ME beat MS and HD
LV and IH both beat HD
By (9): CB JT
By (8): CB JT IH
By (7): CB JT IH LV
By (2): CB JT IH LV ME
By (11): CB JT IH LV ME MS
By (4): CB JT IH LV ME MS LD
By (6): CB JT IH LV ME MS LD JK
By (5): CB JT IH LV ME MS LD JK HD
By (3): CB JT IH LV ME MS LD JK HD JP
Puzzle 251
Which number, when written as a word, makes this true:
Cleverly, there are exactly {?} E's in this sentence.
Puzzle Copyright © Kevin Stone
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Hint
Does ONE work, does TWO work? How many E's are there already?
Answer
Twelve.
"Cleverly, there are exactly twelve E's in this sentence."
Puzzle 252
You have a very large number of generic plastic building blocks, each in the shape of a cube.
Think about how many cubes you would need in order to construct a giant cube with 16 small cubes along each edge.
If you were to then remove the outside layer, how many small cubes would you have removed?
Puzzle Copyright © Kevin Stone
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Hint
How large would the smaller cube be?
Answer
1,352 small cubes.
Reasoning
The original large cube with 16 small cubes along each edge would require 16 x 16 x 16 = 4,096 small cubes.
Removing the outside layer would leave a large cube now with 14 small cubes along each edge (which requires 14 x 14 x 14 = 2,744 small cubes).
It isn't 15 small cubes because you're removing both ends of an edge.
Therefore, you have removed 4,096 – 2,744 = 1,352 small cubes.
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