20 × 20 × 20 solid cubes

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A 20*20*20 solid cube is made of 2000 blocks: 2*2*1 blocks.
It is proved that the cube can be stabbed in a straight line perpendicular to one of its faces,
This line passes through the inside of the cube but does not pass through any brick.

 

Um2002 Grade: Lieutenant Prestige:3 Article: 161 Credit: 7721 Registration: 2002-05-20	4th floor     20 × 20 × 20 solid cubes The square of 6*6 is covered with a rectangle of 1*2. You can find a straight line to divide the square lattice into two parts. Without damaging any rectangle. Is it the same?

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Tycolion Title: Forum Consultant Grade: VIP Prestige:10 Article: 4680 Credit: 51187 Sects: the holy men Heaven and Man sect Registration: 2002-04-04

5th floor     20 × 20 × 20 solid cubes [Quote]The following references are provided by [u] um2002 [/u] InAMPublished content: The square of 6*6 is covered with a rectangle of 1*2. You can find a straight line to divide the square lattice into two parts. Without damaging any rectangle. Is it the same? [/Quote] Are you prompted? :) The beautiful man is just like the autumn of last night; <br> the dust is not enough, and the breeze is blowing away.<Br> [Move]Dark don't move fundamental printing, hand knot King Kong wheel printing, ten fingers Lotus, spiral strength, broken void and .....[/Move] Ashdfsiod <font color = # 00008b> <B> <font size = 2> let's go to the Red Tower to take a boat, just like last night's autumn;

Tycolion Title: Forum Consultant Grade: VIP Prestige:10 Article: 4680 Credit: 51187 Sects: the holy men Heaven and Man sect Registration: 2002-04-04

12th floor     20 × 20 × 20 solid cubes [This post was last edited by tycolion at AM] First, you only need to consider a straight line that goes through an integral point parallel to a plane. There are 19*19*3 = 1083 lines. Then, if a straight line passes through a brick, it must pass through an even number, that is, at least two blocks. The proof is as follows: If there is a straight line that only passes through exactly one brick. Consider the division, that is, the two vertical planes passing through this line divide the entire cube into four areas. Each area of the brick passed by this straight line has 1 small block. Note that a plane parallel to a surface over a whole point either fails to cut the brick or splits the brick into two parts (that is, two small parts each side ). Therefore, apart from the four-part brick, the remaining two planes must be even. Therefore, each region must be an even number + 1, that is, there must be an odd number of small squares. However, it is clear that each region has an even number of small blocks. Conflict! Therefore, if each line passes through a brick, it must pass through at least two blocks. Finally, it is noted that each brick cannot be crossed by two such straight lines at the same time, Therefore, 1083 such lines must be at least 1083*2 = 2166> 2000 (total number of bricks) It is impossible, so the original proposition is true! QED [Img] http://www.iqstar.net/bbs/usr/9/9_902_11.jpg#/img] The beautiful man is just like the autumn of last night; <br> the dust is not enough, and the breeze is blowing away.<Br> [Move]Dark don't move fundamental printing, hand knot King Kong wheel printing, ten fingers Lotus, spiral strength, broken void and .....[/Move] Ashdfsiod <font color = # 00008b> <B> <font size = 2> let's go to the Red Tower to take a boat, just like last night's autumn;