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Charlie has a collection of books that he wishes to display in a narrow bookcase with shelves of width 56 cm.
The thickest books are no more than 16 cm wide and, when placed side by side, the entire collection takes up 2.4m.
Find, with justification, the minimum number of shelves required to guarantee
that all of the books can be displayed in the bookcase.
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I will solve the problem in two steps.
First, I will show that for some special case under the condition, 5 shelves are not enough.
Then I will prove that 6 shelves are enough for any case under the condition.
Let all the books are 15 cm wide.
Then the total number of books is = 16,
and we can place ONLY 3 such books at each shelf.
So, having 5 shelves, we can place only 3*5 = 15 such books, and we need then
the 6-th shelf for the 16-th book.
Now, after completing this case, I can solve the problem in full.
My statement is that 6 shelves is always enough.
1) In the 1-st shelf, I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
2) In the 2-nd shelf, I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
3) In the 3-rd shelf, I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
4) In the 4-th shelf, I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
5) In the 5-th shelf, I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
6) In the 6-th shelf I can fill at least 40 cm of 56 cm.
Indeed, if less than 40 cm is filled, then I can add any book (since it is no
thicker than 16 cm).
So, I can fill at least 40 cm of 56 cm in each of 6 shelves.
Taken together, 6 times 40 cm comprise 2 m 40 cm,
which means that ALL the books will be placed in 6 shelves.
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THE PROOF IS COMPLETED.
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It is a TRUE Math Olympiad level Math problem (!)