## Trigons

 ``` .--A--.--B--. / \ / \ / \ C D E F G H .--I--.--J--.--K--. / \ / \ / \ / \ L M N O P Q R S .--S--.--T--.--U--.--V--. W X Y Z a b c d \ / \ / \ / \ / .--e--.--f--.--g--. h i j k l m \ / \ / \ / .--n--.--o--. ``` Trigons are similar to Triominos, except instead of having numbers at each corner, the numbers are on the sides of each triangular piece. Can you take a set of 24 trigons, consisting of all possible configurations of the values 0, 1, 2, and 3, and place them into a hexagon, two units on a side, such that each adjacent side matches correctly? Or, show why it can't be done. (Note that 1-2-3 is a different trigon than 1-3-2, since neither can be rotated to create the other; while 1-1-2 would be the same as 1-2-1, since the latter can be rotated to obtain the former.)
Since I think the above problem may be easily solved, consider the following extension puzzles:
• Try to solve the puzzle, making the sum at the 7 internal points the same (the sum of the six numbers surrounding the 7 internal points).
• Try to solve the puzzle, making the six internal trigons have different sums. (The sum of the three numbers on each trigon.)

Source: Original, based on a puzzle in Dell's Math Puzzles and Logic Problems magazine.

Henry Bottomley sent a solution which solves the second extension puzzle (different sums for the six internal trigons.) He theorizes that the first extension cannot be solved. (Solution received October 19, 1999.)
```      .--2--.--2--.
/ \   / \   / \
3   3 3   3 1   2
.--0--.--1--.--2--.
/ \   / \   / \   / \
2   0 1   1 2   2 2   0
.--0--.--0--.--3--.--3--.
0   3 1   2 3   3 0   1
\ /   \ /   \ /   \ /
.--1--.--0--.--1--.
3   2 0   0 1   1
\ /   \ /   \ /
.--2--.--2--.
```

Update October 16, 2001. Sandy Thompson sent the following email, showing several solutions to the first extension:
```Yes, Trigons!  From all the way back on August 6, 1999!  Specifically, I'm
addressing the missing solution for the first extension: "Try to solve the
puzzle, making the sum at the 7 internal points the same (the sum of the six
numbers surrounding the 7 internal points)."

And yes, it's Sandy again, after a long hiatus.

After manually searching for a solution for days on end (and coming
agonizingly close a number of times!), I took the cheezy route and created a
rather monstrous program to answer the question for me.

Seeing as one of the pieces in the completed solution must be [3,3,3], the
obvious minimum sum of an internal point (assuming the other four sides
connected to it can be 0) is 6.  Conversely, the maximum sum of an internal
point is 12 (using [0,0,0] and 3's on the other four sides).  A sum of 9 for
an internal point is an important center point, since each side has an
average number of 1.5, and 1.5 * 6 = 9.

After an exhaustive search which took my computer over a week to complete, I
can positively say there are no sum 6 or sum 12 solutions.  I'm sure someone
slick with mathematical logic can prove it somehow... I won't even try.
However, there are TONNES of solutions for sums 7-11.  The program found the
first few solutions for sum 7 and sum 8 within a few hours of completing sum
6.  Then, in a mere 16 minutes, it spit out over 1000 solutions for sum 9...
and it was less than 1% through the set of possibilities when I realized
what was happening and stopped it.

There was no need to check for sums of 10-12, as sum 10 is essentially the
"complement" of sum 8, sum 11 is essentially the "complement" of sum 7, and
sum 12 is essentially the "complement" of sum 6.  All you have to do is take
a completed solution and map all 0's to 3's, all 1's to 2's, all 2's to 1's,
and all 3's to 0's.

And now, the moment you've been waiting for for over two years!

Sum 7:

.-- 3 --.-- 3 --.
/ \     / \     / \
3   2   1   1   1   2
/     \ /     \ /     \
.-- 0 --.-- 1 --.-- 1 --.
/ \     / \     / \     / \
2   1   0   3   3   0   3   3
/     \ /     \ /     \ /     \
.-- 2 --.-- 1 --.-- 0 --.-- 3 --.
\     / \     / \     / \     /
2   2   1   0   0   0   1   2
\ /     \ /     \ /     \ /
.-- 0 --.-- 2 --.-- 2 --.
\     / \     / \     /
3   2   2   0   3   3
\ /     \ /     \ /
.-- 3 --.-- 3 --.

Sum 8:

.-- 1 --.-- 2 --.
/ \     / \     / \
3   0   0   0   1   3
/     \ /     \ /     \
.-- 2 --.-- 3 --.-- 0 --.
/ \     / \     / \     / \
3   3   1   2   3   1   1   2
/     \ /     \ /     \ /     \
.-- 0 --.-- 0 --.-- 3 --.-- 1 --.
\     / \     / \     / \     /
2   2   2   0   0   1   1   3
\ /     \ /     \ /     \ /
.-- 2 --.-- 0 --.-- 1 --.
\     / \     / \     /
3   2   2   3   3   2
\ /     \ /     \ /
.-- 1 --.-- 3 --.

Sum 9:

.-- 2 --.-- 3 --.
/ \     / \     / \
1   0   1   0   0   1
/     \ /     \ /     \
.-- 0 --.-- 3 --.-- 2 --.
/ \     / \     / \     / \
2   2   2   3   3   1   1   1
/     \ /     \ /     \ /     \
.-- 2 --.-- 0 --.-- 3 --.-- 1 --.
\     / \     / \     / \     /
3   1   2   0   0   2   1   0
\ /     \ /     \ /     \ /
.-- 3 --.-- 0 --.-- 2 --.
\     / \     / \     /
1   1   3   3   2   3
\ /     \ /     \ /
.-- 0 --.-- 3 --.

I could send hundreds more, as well as mapped solutions to sum 10 and sum 11
too, but who needs 'em?

As there are so many, I am absolutely astonished that no one found a
solution.  I focused on sum 9 for most of my week or so of manual searching,
and wasn't able to seal the deal.

```

Mail to Ken