- 19 Jan '06 23:49John the Typical Art Student is in a color theory class. his his teacher tells him to make an 8 by 8 grid, and color each cell a singel unique color, using only three colored pencils: red, yellow and blue. she advises that he play about with them a little before hand to get a feel for how they work on the paper. to his desmay he finds that he can only get two consistand shades, light and medium, out of each pencil, which really bums him out because he thought he was being smart when he bought the cheap pencils. he is pretty sure he can not do the assignment with these pencils, untill he finds that when he layers the pencils, each order of application produces a different color. can John compleat his grid?
- 20 Jan '06 00:04

I'm not totally clear on the rules you've stated, but I think John can complete the grid.*Originally posted by fearlessleader***John the Typical Art Student is in a color theory class. his his teacher tells him to make an 8 by 8 grid, and color each cell a singel unique color, using only three colored pencils: red, yellow and blue. she advises that he play about with them a little before hand to get a feel for how they work on the paper. to his desmay he finds that he can only ...[text shortened]... s the pencils, each order of application produces a different color. can John compleat his grid?**

An 8x8 grid requires 64 unique colours. John has 3 coloured pencils, and 2 shades for each colour. Also, the order of application is important. So:

1. Using one pencil at a time, John can create (2 shades) * (3 pencils, choose 1) = 6 colours

2. Using two pencils at a time, John can create (2 shades) * (3 pencils, choose 1) * (2 shades) * (2 pencils, choose 1) = 24 colours

3. Using three pencils at a time, John can create (2 shades) * (3 pencils, choose 1) * (2 shades) * (2 pencils, choose 1) * (2 shades) * (1 pencil, choose 1) = 48 colours

And 6 + 24 + 48 = 78 colours, so John can complete the grid. - 24 Jan '06 09:473 pencils, 2 shades, is equal to 6 colors.

He can use a single pencil on a cell: 6 possibilities

He can use two pencils on a cell: 30 possibilities (including different orders)

He can use three pencils on a cell: 120

Four pencils: 360

Five: 720

Six: 720

He can leave the cell blank: 1

1+720+720+360+120+30+6 = 1957 possible colorisations on a cell.

64 cells, so yeah, he can do it. - 24 Jan '06 10:32

I don't think you can call three pencils with 2 thicknesses each equal to 6 different colours.*Originally posted by TheMaster37***3 pencils, 2 shades, is equal to 6 colors.**

He can use a single pencil on a cell: 6 possibilities

He can use two pencils on a cell: 30 possibilities (including different orders)

He can use three pencils on a cell: 120

Four pencils: 360

Five: 720

Six: 720

He can leave the cell blank: 1

1+720+720+360+120+30+6 = 1957 possible colorisations on a cell.

64 cells, so yeah, he can do it.

R1 + R2 wouldn't be a new shade but it would be R2 wouldn't it? Otherwise he wouldn't have been limited to two shades per pencil in the first place.