1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Kipish [7]
3 years ago
13

Prove that it is impossible to dissect a cube into finitely many cubes, no two of which are the same size.

Mathematics
1 answer:
solniwko [45]3 years ago
3 0

explanation:

The sides of a cube are squares, and they are covered by the respective sides of the cubes covering that side of the big cube. If we can show that a sqaure cannot be descomposed in squares of different sides, then we are done.

We cover the bottom side of that square with the bottom side of smaller squares. Above each square there is at least one square. Those squares have different heights, and they can have more or less (but not equal) height than the square they have below.

There is one square, lets call it A, that has minimum height among the squares that cover the bottom line, a bigger sqaure cannot fit above A because it would overlap with A's neighbours, so the selected square, lets call it B, should have less height than A itself.

There should be a 'hole' between B and at least one of A's neighbours, this hole is a rectangle with height equal to B's height. Since we cant use squares of similar sizes, we need at least 2 squares covering the 'hole', or a big sqaure that will form another hole above B, making this problem inifnite. If we use 2 or more squares, those sqaures height's combined should be at least equal than the height of B. Lets call C the small square that is next to B and above A in the 'hole'. C has even less height than B (otherwise, C would form the 'hole' above B as we described before). There are 2 possibilities:

  • C has similar size than the difference between A and B
  • C has smaller size than the difference between A and B

If the second case would be true, next to C and above A there should be another 'hole', making this problem infinite. Assuming the first case is true, then C would fit perfectly above A and between B and A's neighborhood.  Leaving a small rectangle above it that was part of the original hole.

That small rectangle has base length similar than the sides of C, so it cant be covered by a single square. The small sqaure you would use to cover that rectangle that is above to C and next to B, lets call it D, would leave another 'hole' above C and between D and A's neighborhood.

As you can see, this problem recursively forces you to use smaller and smaller squares, to a never end. You cant cover a sqaure with a finite number of squares and, as a result, you cant cover a cube with finite cubes.

You might be interested in
Find the components of the vector v with initial point P and terminal point Q. Find the unit vector u in the direction of v.
love history [14]

Answer:

Step-by-step explanation:

v=Q-P

v=(5,1,0)

u=v/(magnitude of v)=(5,1,0)/(\sqrt{5^2+1^2+0^2)

u=(5/5.09,1/5.09,0)

v=Q-P

v=(1,1,0)

u=v/(magnitude of v)=(1,1,0)/(\sqrt{1^2+1^2+0^2)

u=(1/\sqrt{2},1/\sqrt{2},0)

8 0
2 years ago
Which statement is NOT always true?
BartSMP [9]
B. The product of two irrational numbers if rational

Example: The square root of 2 times the square root of 3, is the square root of 6, which is still irrational. 
3 0
3 years ago
In math i have to find the percent cost of 6 prices but i'm not sure how to.
tresset_1 [31]
My answer got deleted for some reason, so here we go again. To convert a number to a percent, you multiply the number by 100. So if you had the number 2.4, you would multiply it by 100 to get 240%. After multiplying it by 100, simply add a percent sign, and you're done! 
3 0
3 years ago
The graph of a linear
Gwar [14]

Answer:

Where is the graph?

Step-by-step explanation:

8 0
3 years ago
Write an equation in standard form of a line that passes through (4, 2) and is parallel to y = - 3/4 * x - 5 .
Bingel [31]

Answer:

y=2x+y-int

Step-by-step explanation:

If the line is parallet to the defined by the given equation the slope of the unknown line is m=2.

Use this value of slope to calculate the y intercept. 2  = ( 2 - y-int)/4 - 0)

THus, your equation is y = 2x + y-int

5 0
3 years ago
Other questions:
  • What is the most logical first step in solving the equation x^2+10x+25=11?
    8·1 answer
  • Which of the following would be a good name for the function that takes the pages in a reading assignment and returns the time n
    6·2 answers
  • At a restaurant the cost for a breakfast taco and a small glass of milk is $2.10 the cost for two tacos and three small glasses
    9·2 answers
  • What is the equation of a line parallel to the x-axis that passes though the point (4,5)? What is the slope of this line
    15·1 answer
  • You purchase a new motor scooter for $3,350. You put a 10% down payment and $108.08 per month on a 36 month purchase plan. Use t
    14·1 answer
  • For any positive numbers a, b, and d, with b. 1, logo(ag=
    11·1 answer
  • 3x2 = 81<br> b. (x - 1)(x + 1)-9 = 5x<br> C. X - 9x + 10 = 32<br> d. 6x(x - 9) = 29
    14·1 answer
  • Which answer is it ?
    14·1 answer
  • Molly shot 20 baskets in 4 minutes. Nico shot 42 baskets in 6 minutes. How many more baskets did Nico shoot per day minute?
    9·2 answers
  • Please help ASAP!!!!!!!!!!!!!!!!!!!!!!!!!
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!