Math tapestry puzzle #3

Question

Continuation from Math tapestry puzzle #2

Now things are heating up, it looks like this is slowly becoming bigger. I saw my friend Anna working on her tapestry. She asks me, "Do you like it?"

-- 77 -- -- 06 -- -- 10 --
21 -- AA 51 -- 18 BB -- 58
-- HH -- -- JJ -- -- CC --
-- 76 -- -- 12 -- -- 79 --
17 -- II MM -- 91 KK -- 35
-- 05 -- -- 17 -- -- 40 --
-- GG -- -- LL -- -- DD --
19 -- FF 67 -- 11 EE -- 15
-- 24 -- -- 02 -- -- 07 --

I don't know what happened here, but for sure she's not finished. Can you help her finish her work, given that:

• No variables begin with 6 or 9, nor do they end with 8, or 9.
• AA and CC are prime.
• She has used exactly one number twice.

BONUS QUESTION: In a 6 by 6 puzzle like this, what is the most amount of covered squares allowed and still be able to solve the puzzle?

Answer 1

Step 1:

The first step is to look at some of the adjacent pairs. The largest possible sum is constrained by the $(05,GG)$ pair. The largest value for $GG$ is $87$, making the largest possible sum $05+87=92$. The smallest value for $KK$ is $00$ making the smallest possible sum $91+00=91$ via the $(91,KK)$ pair. Since we know $AA$ is prime and it's in a pair with $51$, it must be $41$ making the pair sum $41+51=92$ (same logic works for $CC$).

Step 2:

Now that we know the pair sum is $92$, we can find the value of all variables except $II$ and $MM$ (since they are paired with each other).
$AA=92-51=41$
$BB=92-18=74$
$HH=92-76=16$
$JJ=92-12=80$
$CC=92-79=13$
$KK=92-91=01$
$GG=92-05=87$
$LL=92-17=75$
$DD=92-40=52$
$FF=92-67=25$
$EE=92-11=81$

Step 3:

We can now fill in any of the 7 completed boxes to get the box sum of $155$:
$06+51+18+JJ=06+51+18+80=155$
$79+35+40+KK=79+35+40+01=155$
$67+11+02+LL=67+11+02+75=155$
$21+77+AA+HH=21+77+41+16=155$
$10+58+BB+CC=10+58+74+13=155$
$19+24+FF+GG=19+24+25+87=155$
$15+07+EE+DD=15+07+81+52=155$

This lets us find the values for $II$ and $MM$:
$II=155-76-17-05=57$
$MM=155-12-91-17=35$
And just to double check $II+MM=57+35=92$

The tapestry:

-- 77 -- -- 06 -- -- 10 --
21 -- 41 51 -- 18 77 -- 58
-- 16 -- -- 80 -- -- 13 --
-- 76 -- -- 12 -- -- 79 --
17 -- 57 35 -- 91 01 -- 35
-- 05 -- -- 17 -- -- 40 --
-- 87 -- -- 75 -- -- 52 --
19 -- 25 67 -- 11 81 -- 15
-- 24 -- -- 02 -- -- 07 --

But wait a second, something is not right...

If you were keen eyed you might have spotted two errors in the final tapestry. First, the rules specify that no variables end with 2. However, $DD$ is supposed to equal $52$. Also, the original rules claim no number is used twice. However, 35 is (once as $MM$ and another time in the unfinished tapestry). There's many ways to fix this, so here's one example of a tapestry that follows all the rules with only changing 4 numbers.

-- 77 -- -- 06 -- -- 10 --
21 -- 41 51 -- 18 77 -- 58
-- 16 -- -- 80 -- -- 13 --
-- 76 -- -- 12 -- -- 79 --
17 -- 57 35 -- 91 01 -- 36
-- 05 -- -- 17 -- -- 39 --
-- 87 -- -- 75 -- -- 53 --
19 -- 25 67 -- 11 81 -- 14
-- 24 -- -- 02 -- -- 07 --