The Lemon Fool

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An artist has divided a non-rectangular canvas into the regions shown, each of which is 8 square feet in area except for the top region which is 16 square feet. He decided to use just four colours and to fill each region with one solid colour in such a way that there would be a different colour on each side of every common border. When he checked his paint supplies he found that he had only

enough red paint to cover 24 square feet
enough yellow paint to cover 24 square feet
enough green paint to cover 16 square feet
enough blue paint to cover 8 square feet

How did he manage to complete his canvas?

Cinelli

cinelli wrote:How did he manage to complete his canvas?

Cinelli

Unless this is a kind of trick question, it looks to me like there is no solution. My reasoning is as follows:

-There is enough paint to cover the entire canvas exactly, thus all the paint needs to be used.
-Only one region is larger than 8 sq ft. Thus between them red and yellow need to fill at least 5 separate regions.
-Thus one of red or yellow needs to fill three separate regions. By inspection, there are not three separate and non-contiguous regions to paint the same colour.

If some regions can be left unpainted a solution is possible, but the problem explicitly stated "He decided [...] to fill each region with one solid colour".

Now, no doubt someone will blow the above out of the water! Bring it on...


GS

Spoiler:

Let one unit of paint be that required to cover one square foot:

Mix 8 units of red paint with 8 units of yellow paint to make 16 units of orange paint.

Mix the green paint with the blue paint to make 24 units of blue-green paint.

This leaves 16 units of red paint and 16 units of yellow paint.

---------------------------------
|          BLUE-GREEN           |
-   -------------------------   -
|   |               |       |   |
-   -    YELLOW     -       -   -
|   |               |       |   |
-   -----------------  RED  -   -
|   |       |       |       |   |
-   -       -       -       -   -
|   |       | BLUE- |       |   |
-----ORANGE - GREEN -------------
|   |       |       |  ORANGE   |
-   -       -       -   ---------
|   |       |       |   |     
-   -----------------   -
|   |               |   |
-   -    YELLOW     -   -
|   |               |   |
-   -----------------   -
|     RED       |       |
-------------------------

Julian F. G. W.

A rendition of it here,

Julian F. G. W.

I reckon it's worth £50 000 at least!

Julian F. G. W.

I don't think you needed to mix the blue and the green, they would have been fine as separate colours.

Scott.

swill453 wrote:I don't think you needed to mix the blue and the green, they would have been fine as separate colours.

cinelli wrote:He decided to use just four colours

I suppose he could have "used" the four colours to paint the canvas in five different colours. That is not how I read the question, however.

Julian F. G. W.

jfgw wrote:

cinelli wrote:He decided to use just four colours

Ah, I see your reasoning. Though not sure I agree it's necessary.

Scott.

Well done Julian for his solution. It’s not quite the same as mine, but that’s a detail. For completeness I painted two panels with red then mixed the remainder with the blue to make purple. So my finished art work looked like this:

---------------------------------
|             Y                 |
-   -------------------------   -
|   |               |       |   |
-   -       R       -       -   -
|   |               |       |   |
-   -----------------   P   -   -
|   |       |       |       |   |
-   -       -       -       -   -
|   |       |       |       |   |
-----   G   -   Y   -------------
|   |       |       |           |
-   -       -       -   ---------
|   |       |       |   |     
-   -----------------   -
|   |               | G |
-   -       R       -   -
|   |               |   |
-   -----------------   -
|     P         |       |
-------------------------

But I take my hat off to Julian for his colourful representation of the finished piece. I have no idea how he did it.

Cinelli

cinelli wrote:But I take my hat off to Julian for his colourful representation of the finished piece. I have no idea how he did it.

Not the neatest way I'm sure (and definitely not the most up-to-date software):
I used Autocad 2000 to draw the pattern. (I'm sure I could have done the whole thing in CAD but I am more familiar with Photoshop.)
Print Preview and press Print Scrn;
Paste into Photoshop 7 and select an area with the magic wand. Expand the selection to get rid of the black line. Select a foreground colour and fill. Repeat for the other areas. Delete the remainder of the black line around the whole pattern.

It made the valves on the motherboard glow a bit but it worked.

Julian F. G. W.

cinelli wrote:Well done Julian for his solution. It’s not quite the same as mine, but that’s a detail.
Cinelli

Once you have the form of a solution, it's all trivial detail. The form is that you need four colours in 3-2-2-2 proportion. Hence for instance, mix the blue with one third of each of the red and yellow to get - I guess - something of a muddy appearance.

For those of us who don't do graphics, can I request that any such puzzles in future come with a label for each region (e.g. 1-8 in this case), to help with posting text-only solutions?

cinelli wrote:But I take my hat off to Julian for his colourful representation of the finished piece. I have no idea how he did it.

Julian has answered about how he prepared the image, and that may be what you had no idea about. But if it's instead about how he got the image into his post, click the " button in his post (as if starting a reply-with-quote) and study the stuff that appears in the input window for your reply. It's quite short and simple, and there is of course no need to complete the reply and submit it!

And by the way, anyone putting an image into a post that way ought to actually say that they drew the image, to reassure the moderators that they own the copyright to it, or say why they otherwise have the right to post copies of it (see the bullet starting "All embedded images" in section 2 of the site rules). Getting into the habit of not doing so will sooner or later result in images vanishing from their posts...

Gengulphus

cinelli wrote:Well done Julian for his solution. It’s not quite the same as mine, but that’s a detail. For completeness I painted two panels with red then mixed the remainder with the blue to make purple.

Cinelli

Image: Julian F. G. W.


I prefer mine Probably worth almost as much though, at least £8!

The procedure is neater in that no extra tubs or washing-up are needed. My solution could have been created by painting the yellow and red areas first, then pouring the yellow into the red tub (for example) to make the orange.

Julian F. G. W.