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For the path graphs the program tries all possibilities for finding a solution by combining smaller solutions. A solution can be extended by adding one at the end. This means that Symple Number for Pn is at most one more than the number for Pn-1. It is also possible to combine two solutions with one or two positions in the middle, meaning that the Symple Number of Pn can be calculated with the Symple Number of Pn1 and Pn2 where n1+n2+1==n or n1+n2+2==n. But in this case, the number of groups in each solution should be taken into account, because these need to be added. The remaining number of steps can be made simultaneously. The maximum of these should be added, and finally one to connect the two solutions. A colleague came up with the formulea 2*sqrt(n) - 2 + 4log(n) as an approximation. The results (as printed by the program) by number of groups, preceded by 2*sqrt(n) - 2 + 4log(n), are:
2: 1.328427 2 3: 2.256583 3 3 4: 3.000000 4 3 5: 3.633100 5 4 5 6: 4.191461 6 4 5 7: 4.695180 7 5 5 6 8: 5.156854 8 5 5 6 9: 5.584963 9 6 6 6 8 10: 5.985519 10 6 6 6 8 11: 6.362965 11 7 6 7 8 9 12: 6.720684 12 7 7 7 8 9 13: 7.061322 13 8 7 7 8 9 10 14: 7.386992 14 8 7 7 8 9 10 15: 7.699412 15 9 8 8 8 9 10 11 16: 8.000000 16 9 8 8 8 9 10 11 17: 8.289943 17 10 8 8 9 9 10 11 13 18: 8.570244 18 10 9 8 9 9 10 11 13 19: 8.841762 19 11 9 9 9 10 10 11 13 14
Calculating the Symple Number for square grid graphs is far more complicated, because there are far more possibilities to split a square grid graph in two parts. To calculate an upper limit, the program only calculates splits along a line cutting the graph in two smaller square grid graphs. Again the number of groups need to be taken into account. Joining two square grid graphs with a Pm graph, requires m steps. Also here the trick of covering two points can be used, if the square grid graph with the largest number of simultaneous steps can be filled at the side. For this reason we define S(n,m,g) as an upper limit for the Symple Number of the graph PnxPm that is calculated by joining two smaller square grid graphs with a Pm graph or extending a single square grid graph with such a graph. For such a graph it is true that either a vertex at the top or at the bottom can be filled as a last step. Which means that a rotated form can be used as a candidate for reduction for a larger square grid graph. The program produces the following table for S(n,m,g) where the minimum Symple Number and the smallest number of groups for which it occurs are given:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1: 1, 1 2, 1 3, 1 3, 2 4, 2 4, 2 5, 2 5, 2 6, 2 6, 2 6, 3 7, 2 7, 3 7, 3 8, 3 8, 3 8, 3 8, 4 9, 3 2: 2, 1 4, 1 5, 2 6, 2 7, 2 8, 2 9, 2 10, 2 11, 2 12, 2 13, 2 13, 3 14, 3 15, 3 15, 3 16, 3 16, 4 17, 3 17, 4 3: 3, 1 6, 1 7, 2 8, 3 10, 2 11, 3 12, 3 13, 3 14, 4 16, 3 17, 4 19, 3 19, 3 20, 4 20, 4 21, 5 22, 4 23, 4 23, 4 4: 3, 2 7, 2 9, 2 10, 3 12, 4 13, 5 15, 4 16, 4 17, 5 18, 5 19, 5 20, 6 21, 6 22, 6 23, 6 25, 5 26, 5 27, 6 27, 6 5: 4, 2 9, 2 11, 2 12, 3 14, 4 16, 3 17, 4 19, 4 20, 4 21, 5 22, 6 24, 6 25, 8 28, 5 29, 6 31, 4 32, 4 32, 5 33, 6 6: 4, 2 10, 2 12, 4 14, 3 16, 4 18, 5 20, 4 21, 5 23, 5 24, 5 25, 6 27, 6 28, 6 29, 8 31, 6 32,11 34, 6 36, 6 37, 6 7: 5, 2 12, 2 14, 4 16, 3 18, 4 20, 5 22, 4 24, 5 25, 6 27, 5 28, 6 29, 7 31, 6 32, 7 33, 8 35, 7 36, 8 40, 7 41, 8 8: 5, 2 13, 2 15, 4 18, 3 20, 4 22, 5 24, 6 26, 5 28, 6 29, 7 31, 8 32, 9 34, 8 35, 9 37, 8 38, 9 39,10 42, 9 43,10 9: 6, 2 15, 2 17, 4 20, 3 22, 4 24, 6 27, 4 29, 6 30, 6 32, 7 33, 8 35, 7 37, 7 38, 8 39, 8 41,10 42,10 44, 8 45, 8 10: 6, 2 16, 2 18, 4 22, 3 24, 4 27, 5 29, 6 31, 6 33, 6 34, 9 36, 8 37, 9 39, 8 41, 9 42,10 44, 9 46, 8 47,10 48,10 11: 6, 3 17, 3 20, 4 24, 3 26, 4 29, 6 31, 7 33, 6 35, 6 37, 7 38, 8 40, 9 42, 9 43,11 45, 8 47,10 48,10 50,11 51,12 12: 7, 2 19, 2 21, 4 26, 3 28, 4 32, 5 33, 8 36, 6 38, 6 40, 7 41, 8 43,10 44,11 46,13 48,10 50, 9 51,10 53,11 54,12 13: 7, 3 20, 3 23, 4 28, 3 30, 4 33, 5 35, 6 37, 8 40, 6 42, 7 43, 8 45,10 47,10 49, 9 50,10 52,12 54,10 55,12 57,10 14: 7, 3 21, 3 24, 4 30, 3 32, 4 35, 6 37, 8 40, 7 42, 8 45, 8 46,12 48,10 50, 8 51,13 53,10 55,12 56,13 58,15 60,12 15: 8, 3 23, 3 26, 4 31, 5 33, 6 36, 6 39, 8 41,10 44, 8 47, 7 49, 8 50,10 52,10 54,11 55,12 57,14 59,10 61,12 62,12 16: 8, 3 24, 3 27, 4 33, 5 35, 6 39, 5 41, 6 43, 8 46,10 49, 7 51, 8 53,10 54,11 56,13 58,14 60,16 62,12 64,14 65,12 17: 8, 3 25, 3 28, 6 34, 6 37, 6 40, 6 43, 8 46, 8 48,10 52, 7 53, 8 55,10 57,12 59,11 61,12 63,14 65,11 66,13 67,14 18: 8, 4 26, 4 30, 4 36, 5 38, 6 42, 6 45, 8 48, 9 51, 8 53, 9 55,10 58,10 60,12 63,13 65,10 66,12 67,13 69,15 70,16 19: 9, 3 28, 3 31, 6 37, 6 40, 6 44, 6 46, 8 49, 9 52, 8 55,10 58, 8 60,10 62,12 65,10 67,11 68,12 69,14 71,15 72,16
An interesting case is 16,6 where the minimum number
of groups, 11, is larger than all it surrounding
values. There are more such cases, but this seems to
the largest in the table. Notice that there are also
cases that are only surrounded by larger numbers,
such as 13,14. The break down of S(16,6,11) based
on output printed by the program is:
Which could be made graphical in the following
figure, where '#' is used for groups, '-' and
'|' for the connecting path graphs, and '+' the
points that act as a reduction point.
That this can be filled in 32 steps is shown in
the following, where the steps are marked with
'a' to 'z' and 'A' to 'F':
###|###|###|+###
###|###|###|####
-------|+-------
##+####|###+####
##|+---|---|+---
##|####|###|####
almqblmCclmrrdlm
noprnopBnopsnopq
uvtwxyzAAuvtwxyz
eltflmnDglmthlmn
mnrropqEnoprropq
opsilmnFjlmsklmn
Wednesday, October 27, 2010
c.2272delG
Today, we received a letter informing us that a c.2272delG
mutation has been found in the MLL2 gen/enzyme
of Andy. It is a new mutation because it is not found with
Li-Xia or me. Of course, it could be possible that either
of us caries a cell line in our sex cells that has this
mutation. The mutation consists of a single deletion of a
base pair at position 2272. To see the consequences for
this appearent small mutation in the amino acid sequence,
look here.
Although, the rules of Symple are rather simple, I feel that they still could be made simpler. One of the problems when playing the game (without computer support) is that you often forget which groups you already have grown. And also the rule that if two groups grow on a shared field that it counts as if both groups have grown, is rather complicated. What would happen if all restrictions on where you can grow are removed? Simply stating that you can grow as many stones as you have groups. That means that if you want to grow in a certain move, you simply take a number of stones equal to the number groups you have. Would this game, that I would brand Super Sympe, be substantial different from Symple? And how different would it be? Whether invasions at the end of the game are possible, depends on the number of groups the opponent has. Interesting.