# Find the missing elephants

Determine color and direction of the missing elephants in the picture.

Here's a textual representation of the above.

G< R< R< G< R< M> G< R< R> M<
G< G< G> G< G> G< G< G< G> G>
G< M> M> G< G< ?? M> M> G< G<
Y< Y> G> G< Y< Y> Y< G> G< Y>
G< G< G> G> M< M> G> G> M< M<
R> M> ?? G< R> M< Y< Y> G> R>
R> R< R< R< R> R> R< ?? R> R<
R< G< R< G< R< G> G> R> G> R<
Y> Y> Y< Y< Y> Y< Y> Y< Y< Y<
Y> Y< Y< Y< Y< Y> Y> Y< Y> Y<

• R: red
• Y: yellow
• G: green
• M: magenta
• >: facing right
• <: facing left
• ?: gap

Hint 1:

The elephant herd has no boundaries. The right side joins the left side and the bottom joins the top.

Hint 2:

Elephants are social animals: they count on themselves and on their neighbours.

Hint 3:

With more fellows around, an elephant must see further.

Hint 4:

Eight are the neighbours. One is the self. Sight overflows. Where is the match?

Hint 5:

Almost everything in this puzzle is random.

• Interesting. My brain is burning out trying to figure it out. :( Nov 26, 2014 at 8:25
• Elephantastic puzzle, this. Nov 26, 2014 at 8:51
• Can this puzzle be solved without all the elephants, or are they all relephant?
– Joe
Nov 26, 2014 at 8:53
• Are R, Y, G, M the only allowed colours, or could there be e.g. blue or cyan elephants too? Nov 26, 2014 at 10:35
• I am very sure this is not the answer,but I just had an idea: Elephants are heavy and can't fly, so the should fall down. Therefore, the three holes should be filled by the elephants above them. :-) I guess I have to continue searching for a real answer....? Nov 26, 2014 at 19:34

OK. I finally figured out this accurs slightly frustrating puzzle. I shall forever curse the name of GOTO for more than an hour spent convolving kernels with elephant herds in vain attempts to find some sort of linear invariant.

The missing cells, in standard reading order, are

green elephant facing right
yellow elephant facing left
red elephant facing left

The rationale is as follows:

An elephant's "gaze" must be fixed on an elephant of identical colour. Let $D$ be the number of elephants in the 3x3 cell block surrounding an elephant that share the elephant's direction (this includes the elephant him/herself). The grid is toroidal, hence the grid "wraps around" at the edges, from right to left, top to bottom. An elephant's gaze is fixed on the cell $D$ cells ahead of him/her in the direction (s)he's looking (respecting wrapping). We shall call this property "elephanticity".

The three colours and directions listed in the above spoiler are the only three combinations that yield elephanticity for all cells in the rows in which the empty cells appear as well as the rows immediately above and below them.

• I love that definition of elephanticity. Are you sure the yellow elephant is facing right? Otherwise this solution is correct. Nov 28, 2014 at 5:55
• @GOTO0 - Ah. Fixed the mistake. I actually had the right answer on paper, but using arrows rather than words, and let's just say I got rather turned around in the translation. :P
– COTO
Nov 28, 2014 at 7:41
• Good job. If you want, you can add this picture showing the solution to your answer, and I shall delete this comment. Nov 28, 2014 at 7:53
• It seems to me the 1st elephant of row 3 doesn't have elephanticity. How do you choose who should have elephanticity? Nov 28, 2014 at 8:32
• @florianf that elephant has 4 friends of same direction hence gazes 5 fields to the left, which is the hole. And the hole has yo be a green elephant. So all is well, isn't it? Nov 28, 2014 at 9:03

I have an idea of a solution which would currently give me for the three elephants (top to bottom): Green,right Green,left Green,right

Reasoning given below in the spoiler. However, my solution is currently flawed, so it is rather the idea I want to post here for commenting.

The puzzle could be turned into a system of equations by assigning the color to a variable name and the direction to +-. Each line would translate into one equation, but the right hand side of the equation is not defined, hence I set it to zero. (Also line 4 seems to indicate this.) This gives me the following, flawed, equation system:

this gives me:

0 = -3 g -3 r ; 0 = -2 g ; 0 = -5 g + 4 v + X1 ; 0 = 0 ; 0 = 2 g - 2 v ; 0 = 3 r + x2 ; 0 = -3 r + x3 ; 0 = g - 3 r ; 0 = -2 y ; 0 = -2 y ;

And solving this from bottom up give my solution.

But the equation system as whole is flawed, so I either made a mistake, or an incorrect assumption. I could try different ways of turning this into equations ( I.e going to vertically, or assuming other right-hand sides, but if I am on the completely wrong track here, I'd rather know it....) other may pick up the idea here as well.

Am I barking up the wrong tree, or just haven't done it properly yet?

• Well, this is maybe simpler than you imagine. Please check out the hints :-) Nov 26, 2014 at 20:50
• I posted this before you have had the hints.Now I see it doesn't fit. Nov 26, 2014 at 21:01
• @GOTO0 Looking at the answer: you still stick to "it is simpler?" ... Don't think so! Very nice puzzle, though Nov 30, 2014 at 21:35
• I should have made it clear from the beginning that the solution could be found by just looking at the board and counting on two hands, and that no advanced mathematical skills were required. This is basically what I meant to say when I wrote simpler, but now I realize that my wording was too vague. Anyway, it's nice to hear that you enjoyed the puzzle. Dec 1, 2014 at 13:44
• @GOTO0 Your wording was good, no problems there. I just think your answer is actually more complex than my attempted one :c) BTW, I've just asked a general puzzle-building questions. I would be very much interested in your answer there. Dec 1, 2014 at 14:19

My guess is, from top to bottom ?? we have

M>, G>, R<

Non-spoiler paragraph would read that I tried to match the pattern of colors vs the left/right ratio between the rows and the columns of the table.

First

It makes the row read 5,0,0,5 for colors and 5,5 for L/R. It affects the column to go 6/4 L/R because we've already matched the 3/7 with the second elephant. It affects column colors to 2,1,3,4

Second

To go 3/7 R/L would match the final column's 3/7 with 2,3,3,2

Third

All rows the same color are 6/4 L/R, so match the pattern

This is mostly guess-work, but you can see my (convoluted) work with this image:

• Some of your guesses look far-fetched, but +1 for analysis. Nov 26, 2014 at 14:01
• @GOTO0 Yeah, the first elephant is where I struggled the most, wish I had a more solid analysis Nov 26, 2014 at 14:02