Lesson Math Olympiad level problem on placing books in bookcase

This Lesson (Math Olympiad level problem on placing books in bookcase) was created by by ikleyn(52776)

: View Source, Show
About ikleyn:

Math Olympiad level problem on placing books in bookcase

Problem 1

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 2.4 m.
Find the minimum number of shelves required to guarantee that all of the books can be displayed in the bookcase.

Solution

I will show first that 5 shelves are not enough.
To demonstrate it, I will construct a contradictory example.

Let all the books are 15 cm wide.


Then the total number of books is   = 16,

and we can place  ONLY  3  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 counter-example, I am ready to 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.

My other additional lessons on Miscellaneous word problems in this site are
    - I do not have enough savings now
    - In a jar, all but 6 are red marbles
    - How many boys and how many girls are there in a family ?
    - What is the last digit of the number a^n ?
    - Find the last three digits of these numbers
    - What are the last two digits of the number 3^123 + 7^123 + 9^123 ?
    - Advanced logical problems
    - Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b) and sqrt(c)
    - Calculus optimization problems for shapes in 2D plane
    - Calculus optimization problems for 3D shapes
    - Solving some linear minimax problems in 3D space
    - Solving one non-linear minimax problems in 3D space
    - Solving linear minimax problem in three unknowns by the simplex method
    - The "pigeonhole principle" problems
    - In the worst case
    - Page numbers on the left and right facing pages of an opened book
    - Selected problems on counting elements in subsets of a given finite set
    - How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ?
    - Nice problems to setup them using Venn diagram
    - Wrapping a gift
    - In preparation for Halloween
    - Nice entertainment problems related to divisibility property
    - Stars and bars method for Combinatorics problems
    - Math Olympiad level problem on caves and bats
    - Math Olympiad level problem on caught fishes
    - Math Olympiad level problem on pigeonhole principle
    - OVERVIEW of additional lessons on Miscellaneous word problems

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

This lesson has been accessed 938 times.