I think you had found the solution (756;792;800).
I've found another solution, i think is the minimal (768;770;810), dif 42
JMB

I think that your solution is 12! = 756×792×800.
My better solution is 12! = 768×770×810. My difference is 42.
I know that better solution doesn't exist.
Slawomir Wojcik

Yesterday I sent solution for 12! After that I thought a little, wrote a
program and...
Here a solution for "Factorials"
10!=140*160*162=3628800 (22)
11!=324*350*352=39916800 (28)
12!=768*770*810=479001600 (42)
13!=1800*1848*1872=6227020800 (72)
14!=4368*4455*4480=87178291200 (112)
15!=10800*10920*11088=1307674368000 (288)
I'm not sure that these solutions are the best ones for N>15...
For N<=15 I use "Brutal Force" Algorithm, so the solutions are correct.
But for bigger N I wrote other euristic algorithm.
May be anyone can find better results.
16!=27300*27648*27720=20922789888000 (420)
17!=70560*70720*71280=355687428096000 (720)
18!=184800*185895*186368=6402373705728000 (1568)
19!=494208*496125*496128=121645100408832000 (1920)
20!=1344000*1343680*1347192=2432902008176640000 (3512)
Is it nessesary to continue?
Best Regards, NorT (Eugene V. Bryzgalov)

Your product 756*792*800 can be bettered only 768*770*810, a difference of 42.
It made me wonder about what happens more than three numbers are allowed.
Regards
Mark Michell

Yours must have been 756, 792, 800.  Slightly better with a difference 
of 42 is 768, 770, 810.
There are exactly 6 possible divisors in the 740 to 820 range, and their 
product is the SQUARE of 12 factorial, so if 3 multiply out, then so do 
the other 3.  Wonder if this could happen again?
Gary Mulkey

Trial and Error, using a forcing algorithm, gave me this result.  Would not be suprised if there are better, because you can hit a wall with this algo.
 1 16 15 4 7 10 20   = 1,344,000
9 11 12 6 2 19 5     = 1,354,320
13 17 18 14 3 8      = 1,336,608
difference                      17,712
However, it is still pretty good and I would like to see others.
Dick Saunders Jr.

I think I've surprised myself by finding the definitive solution to the
problem of how to divide the first 25 primes into three packages to
obtain the closest products.
1321040331310   {2, 5, 7, 19, 43, 53, 59, 83, 89}
1321053487611   {3, 11, 13, 17, 23, 41, 47, 61, 67}
1321117514527   {29, 31, 37, 71, 73, 79, 97}
Difference = 77183217 
For your information, my method was as follows:
1)  I used a simple genetic algorithm to evolve an approximate
solution.
2)  I used the solution from (1) to limit the search space of a
brute force search for sets of 7, 8, 9 or 10 primes with products
near the required total.  I had only 441 such sets, so you can
see how much this limited the search space.
3)  Another brute force search of these 441 sets found 252
potential solutions (i.e. picking 3 non-overlapping sets), and it
was easy to find the best of these.
Hopefully, I didn't make any mistakes along the way, meaning the
above solution is optimal.
I've also found 768 x 770 x 810 = 12! having a difference of only 42.
I think this is optimal.
Finally, I have 1343034 x 1344000 x 1347840 = 20!
This has a difference of 4806 but I haven't done enough yet to
confirm if this is optimal.
Thanks for a great set of problems recently.
Regards
Nick Gardner

In my opinion very easy (with excel in 4 minutes)
12! = 768*770*810 ( max dif = 42)
Thanks for your puzzle
Andre Wauters

I think I can beat your split of 12!
12! = 810 * 770 * 768, and 810 - 768 = 42
I agree with your answers for 10! and 11!.
Jon Palin

Here are my best solutions. I think my search is exhaustive, but since I
didn't use infinite precision arithmetic I'm still a bit worried about the
correctness (i.e. these numbers being the actual minima)
2*3*4=4! Diff=2
4*5*6=5! Diff=2
8*9*10=6! Diff=2
14*18*20=7! Diff=6 AND 15*16*21=7! Diff=6
32*35*36=8! Diff=4
63*72*80=9! Diff=17  AND 64*70*81=9! Diff=17
140*160*162=10! Diff=22
324*350*352=11! Diff=28
768*770*810 = 12! Diff=42
1800*1848*1872=13! Diff=72
4368*4455*4480=14! Diff=112
10800*10920*11088=15! Diff=288
27300*27648*27720=16! Diff=420
70560*70720*71280=17! Diff=720 (=6!)
184800*185895*186368=18! Diff=1568
494208*496125*496128=19! Diff=1920
1343680*1344000*1347192=20! Diff=3512
3704400*3706560*3720960=21! Diff=16560
Greetings,
Luc Kumps

Here are my results for the factorial cake problem.
Exhaustive search
3!   2    1x2x3
4!   2    2x3x4
5!   2    4x5x6
6!   2    8x9x10
7!   6    14x18x20
8!   4    32x35x36
9!   2    70x72x72       or    17    63x72x80
10!  22   140x160x62
11!  28   324x350x352
12!  42   768x770x810
13!  72   1800x1848x1872
Search around n!^1/3
14!  112  4368x4455x4480
15!  288  10800x10920x11088
16!  420  27300x27648x27720
17!  720  70560x70720x71280
18!  1568 184800x185895x186368
19!  1920 494208x496125x496128
20!
Cheers!
Jim Shaw

768 x 770 x 810 = 12!
(difference = 42)
Regards,
Igor Krivokon

1 * 10 * 7 * 11  = 770
9 *  3 * 6 *  5  = 810
2 *  4 * 8 * 12  = 768
For a difference of 42.
Dick Saunders Jr.

768 * 770 * 810 = 12! (810 - 768 = 42)
Denis Borris Ottawa Ontario Canada.

Hi Ed,
12! = 768*770*810  (diff.42)
Best puzzling !
Dario Uri

I think the subject has a difference of 42..... 
Correct me if I am wrong!
P.S. Found by using an excel spreadsheet, and breaking 12!
down into factors in three columns.  Then moving the factors
around till they looked right.
Adam Dewbery

I managed to find your solution.  It wouldn't seem to had to write a 
computer program to check all the possibilities.  12! has only(?) 792 
factors.
I'll keep looking.  Maybe I'll try going higher or lower.
-Jeremy Galvagni

How about:
12.8.4.2=768
11.10.7.1=770
9.6.5.3=810 =768+42
Regards
Chris Lusby Taylor