I get idea from the emperor's new groove video game.
https://www.youtube.com/watch?v=DtClbzUd4QI#t=160 At 2.40, he go to about 3-levels height.
The question is if Kuzco want to go to 10-levels height, what is minimal pots he needed ?
Let me explain the rule. I have to change some thing to make it easier. In the game, all these rule isn't 100% true. (about 95%)
- Let Kuzco's position and pots' position are integer.
- Let Kuzco can only walk, jump, use his head to put a pot on his back and carry it.
- Let pots are 1x1x1 cube shape.
- There is no space restriction. You can even use [Integer] x [Integer] as space if you want and 10-levels height position can be any where you want in your space.
Walk
- Kuzco can walk 8-direction.
- Kuzco can walk and fall from higher levels to lower levels, no matter different height.
- Kuzco can walk in diagonal direction if there is no pots that block the way, for example,
Number in each cell represent number of pots and height. K is where Kuzco is. Let Kuzco is at (0,0,0). Green cells is where he can walk from (0,0,0) directly. He can't walk to (1,1,0) and (-1,1,0) directly. He has to walk to (1,0,0) and (-1,0,0) before.
Jump
- Kuzco can jump 8-direction.
- Kuzco can jump to max 3 unit distance to same or lower levels.
Let Kuzco is at (0,0,0). Blue cells is where he can jump from (0,0,0) directly.
- Kuzco can jump to max 1.5 unit distance to +1 levels.
- Kuzco can't jump to +2 levels.
- Kuzco can jump in diagonal direction if there is no pots that block the way, for example,
Let Kuzco is at (0,0,0). Blue cells is where he can jump from (0,0,0) directly. He can't jump to (1,-1,0) and (-1,-1,0) from (0,0,0) directly because a pots at (0,-1,0) block the way. There are special cases that he can jump to (2,-1,0) and (-2,-1,0) from (0,0,0) directly in this example.
- Kuzco can't jump while he carry a pot.
Pot
- Kuzco can take a pot and put it on his back 8-direction.
- Kuzco can take a pot only uppermost pot and from only same level and +1 level.
- Kuzco can take a pot at max 1.5 unit distance in both from same level and +1 level.
- Kuzco can take a pot in diagonal direction if there is no pots that block the way, for example,
Let Kuzco is at (0,0,0). Orange cells is where he can take a pot from (0,0,0) directly. He can take pots at (-1,0,0) and (1,1,1) from (0,0,0) directly (A pot on floor is at z=0 and has 1 unit heigh). He can't take pots at (-1,1,0) and (-1,-1,1) from (0,0,0) directly because a pots at (-1,0,0) block the way.
- If Kuzco can take a pot, he can put that pot back to that place it from.
K* is Kuzco with a pot. Let Kuzco is at (0,0,0). Yellow cells is where he can put a pot back from (0,0,0) directly. He can put a pot back to (1,0,0) , (0,1,0) , (0,-1,0) , (-1,0,1) and (1,1,1) from (0,0,0) directly. He can't take pots at (-1,1,1) and (-1,-1,1) from (0,0,0) directly because a pots at (-1,0,0) block the way.
I know that if he want to go to 10-levels high, he needs 9-levels high pots stack. If he want to go to 9-levels high, he needs 8-levels high pots stack. So, the minimal pots he needed is at least 1+2+3+4+5+6+7+8+9=45 pots.
But I don't know if he want to built 9-levels high pots, what is minimal pots he needed ?
This is how to play Kuzco go to 4-levels height with 6 pots.
First Kuzco can make 1-levels and 2-levels high pots stack from the floor(z=0). Then he has to jump to 1-levels high pots stack to move a pots in 2-levels high pots stack to another 2-levels high pots stack to create 3-levels high pots stack. Now he can make ladder pots stack to go to 4-levels height.