topic Re: How to extract submatrix? in SAS/IML Software and Matrix Computations

How to extract submatrix?

zana — Thu, 30 Oct 2014 08:01:25 GMT

Hello,

Currently i have problem for convert some submatrix from an output solution file. The one of output table (this output file contains 4000 table like as follows) is represented as the follows (7*17):

Please let me i show that as a simple way.

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

Now, i would like to create three submatrix. But this is more important, these submatrix could have a different dimensions. For example, maybe the first submatrix have 8*8, the 2nd one 4*4 and the 3rd one have 5*5 (or any other modes).

1st submatrix;

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

2nd submatrix;

37	38	39	40
38	41	42	43
39	42	44	45
40	43	45	46

3rd submatrix;

47	48	49	50	51
48	52	53	54	55
49	53	56	57	58
50	54	57	59	60
51	55	58	60	61

How could i do it?

Many thanks!

Re: How to extract submatrix?

Ksharp — Thu, 30 Oct 2014 10:20:57 GMT

That is not submatrix. That is creating a whole new matrix . The parent matrix only has 17*7 . How can you subset it to be 8*8 ?

Re: How to extract submatrix?

CTorres — Thu, 30 Oct 2014 12:26:04 GMT

Zana,

I think you have to better define the problem to solve:

Having an initial matriz n*m, how do you determine how many submatrixes you need and the dimensions of them?

Please explain.

CTorres

Re: How to extract submatrix?

Rick_SAS — Thu, 30 Oct 2014 12:53:15 GMT

I don't fully understand how you are creating these matrices. It looks like the first matrix is the elements 1:36 packed into a symmetric matrix. The second is 37:46, packed into a symmetric matrix, and the last is 47:61.

Matrices in IML are stored rowwise. You can use subscripts to extract submatrices. The following statements extract the elements from the big matrix and use the SQRVECH function to pack those elements into a symmetric matrix:

proc iml;
A = shape(1:(7*17), 7); /* create 7 x 17 example */

M = A[1:36]; /* extract elements 1:36 */
M1 = sqrvech(M);
print M1;

M = A[37:46]; /* extract elements 37:46 */
M2 = sqrvech(M);
print M2;

Re: How to extract submatrix?

CTorres — Thu, 30 Oct 2014 13:48:41 GMT

Zana,

The following macro includes the logic to generate the submatrixes:

593 %macro Submatrix(initval,dim);
594 data _Null_;
595    array SubM(&dim,&dim);
596    n=&initval;
597    do i=1 to &dim;
598       do j=1 to &dim;
599          if i Le j then do;
600             SubM(i,j)=n;
601             n+1;
602          end;
603          else SubM(i,j)=SubM(j,i);
604       end;
605    end;
606    do i=1 to &dim;
607       do j=1 to &dim;
608          put SubM(i,j) 3.0 @;
609       end;
610       put;
611    end;
612 run;
613 %mend Submatrix;
614 %submatrix(1,8)

1 2 3 4 5 6 7 8
2 9 10 11 12 13 14 15
3 10 16 17 18 19 20 21
4 11 17 22 23 24 25 26
5 12 18 23 27 28 29 30
6 13 19 24 28 31 32 33
7 14 20 25 29 32 34 35
8 15 21 26 30 33 35 36
NOTE: DATA statement used (Total process time):
real time 0.00 seconds
cpu time 0.00 seconds

615 %submatrix(37,4)

37 38 39 40
38 41 42 43
39 42 44 45
40 43 45 46
NOTE: DATA statement used (Total process time):
real time 0.00 seconds
cpu time 0.00 seconds

616 %submatrix(47,5)

47 48 49 50 51
48 52 53 54 55
49 53 56 57 58
50 54 57 59 60
51 55 58 60 61
NOTE: DATA statement used (Total process time):
real time 0.00 seconds
cpu time 0.00 seconds

CTorres

Re: How to extract submatrix?

Rick_SAS — Thu, 30 Oct 2014 14:05:38 GMT

Here it is in SAS/IML:

proc iml;
start Submatrix(A, initval,dim);
   L = dim*(dim+1)/2;
   k = initval:(initval+L-1);
   return( sqrvech(A) );
finish;

A = shape(1:(7*17), 7); /* create 7 x 17 example */

M1 = Submatrix(A, 1, 8);
print M1;
M2 = Submatrix(A, 37, 4);
print M2;
M3 = Submatrix(A, 47, 5);
print M3;

Re: How to extract submatrix?

zana — Thu, 30 Oct 2014 14:51:35 GMT

Dear all thanks for all your reply.

I'm so sorry for delay getting back to you. My request was a little confusing, so i decide to discuss step by step. However, I confused too.

Overall, in some of these submatrix (presented above) included other submatrix (e.g., the submatrix 4*4 had 4 matrix 2*2; this is 2 matrix 2*2 about submatrix 4*4). I think that after extract these main submatrix (8*8, 4*4 & 5*5) maybe it will not so difficult for extracting submatrix in each of these main submatix.

However, it will so well if we could extract these submatix directly.

For clarity, overall my request is as follows:

My table:

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1st submatrix (1st main_submatrix);

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1st submatrix (this can be extracted from 1st main_submatrix [8*8] or directly from my table).

1	5
5	27

2nd submatrix ....

9	13
13	31

3rd submatrix ...

16	20
20	34

4st submatrix ...

22	26
26	36

About 2nd main_submatrix the included submatrix should be like as follows (this can also be extracted from 2nd main_submatrix [4*4] or directly from my table) :

1st ..

37	39
39	44

2nd ..

41	43
43	46

And the last main_submatrix (5*5) do not have any submatrix.

Thank you so much for your kind.

Re: How to extract submatrix?

zana — Thu, 30 Oct 2014 15:00:47 GMT

Dear Rick, thanks for you. I should inform you that this table only is a simple example (my output file is't like that). Please see the original output file (Meanwhile i used SAS9.1.3):

2.826 0.1117 0.5655E-01 -0.2051 -0.9951 -0.3791E-01 -0.2002E-01

-0.7701E-01 0.4744E-02 0.2165E-02 -0.8538E-02 -0.3539E-01 -0.1632E-02 -0.6954E-03

-0.4663E-02 0.1381E-02 -0.1599E-02 -0.2514E-01 -0.9058E-03 -0.4986E-03 0.5249E-03

0.4369E-01 -0.3276E-02 -0.2731E-03 -0.6431E-03 0.2844E-01 1.032 0.2738E-01

0.2465E-01 -0.4330E-01 0.1349E-02 0.7079E-03 -0.1114E-02 0.8279E-03 -0.1147E-02

0.2654E-01 11.59 0.2370 0.2432 0.1908 1.796 0.4569E-01

0.5911E-01 -0.2752 0.1163E-01 0.6655E-02 0.1226E-01 0.1842E-01 0.1013E-02

0.1045E-02 -0.5404E-02 0.7722E-02 0.8686E-02 0.4204E-01 0.1071E-02 0.1562E-02

-0.4825E-02 0.1291 -0.7895E-01 -0.3190E-02 -0.1356E-02 0.9782E-02 1.391

0.3897E-01 0.4140E-01 -0.1326 0.1479E-02 0.1124E-02 -0.3741E-02 0.1285E-02

-0.3616E-02 0.2953E-01 20.56 0.5560 0.5731 -0.6843 0.7305

0.2406E-01 0.4680E-01 0.3018E-01 0.1988E-01 0.1599E-01 -0.2887E-01 0.2563E-01

-0.4890E-03 0.1693E-02 -0.3787E-02 0.1610E-01 -0.1922E-01 0.2705E-01 0.6995E-03

0.1439E-02 -0.2149E-02 0.6893 -0.2031 -0.8243E-02 -0.7854E-02 0.8168E-01

4.398 0.1241 0.1144 -0.1733 0.4284E-02 0.3585E-02 -0.5150E-02

0.3513E-02 -0.3947E-02 0.1259 12.93 0.3446 0.3637 -0.4216

0.4785E-01 0.1194E-01 0.1851E-01 0.1536E-01 -0.3895E-02 1.157

Best regards

Re: How to extract submatrix?

Rick_SAS — Thu, 30 Oct 2014 15:03:35 GMT

You should know that SAS 9.1.3 is more than 10 years old. See How old is your version of SAS? Release dates for SAS software - The DO Loop

Re: How to extract submatrix?

Rick_SAS — Thu, 30 Oct 2014 15:07:39 GMT

You can use subscripts to extract the submatrices:

S1 = M1[{1 5}, {1 5}];

S2 = M1[{2 6}, {2 6}];

S3 = M1[{3 7}, {3 7}];

etc.

Of course, you could put this logic into a loop.

Good luck!

Re: How to extract submatrix?

zana — Thu, 30 Oct 2014 22:32:48 GMT

Dear Torres, so thanks. Please say me how can i run your macro code? (i used SAS 9.1.3)

Thank you.

Re: How to extract submatrix?

CTorres — Fri, 31 Oct 2014 13:01:02 GMT

The following is the macro code. I am sure it will work in SAS 9.1.3:

%macro Submatrix(initval,dim);

data _Null_;

array SubM(&dim,&dim);

n=&initval;

do i=1 to &dim;

do j=1 to &dim;

if i Le j then do;

SubM(i,j)=n;

n+1;

end;

else SubM(i,j)=SubM(j,i);

end;

do i=1 to &dim;

do j=1 to &dim;

put SubM(i,j) 3.0 @;

end;

put;

end;

run;

%mend Submatrix;

The way to run the macro is:

%submatrix(1,8)

%submatrix(37,4)

%submatrix(47,5)

where the first argument is the initial value used to populate the matrix and the second is the dimension of the matrix.

You can include the macro (relevant logic in bold) in your programs.

CTorres

Re: How to extract submatrix?

zana — Fri, 31 Oct 2014 20:31:51 GMT

Thanks Torres. The program for extract submatrix from my table (example) is true. But when i tried to run the program at the following, i can not found my expected output. Please see the program and outputs;

data have;

input v1-v7;

cards;

2.826 0.1117 0.5655E-01 -0.2051 -0.9951 -0.3791E-01 -0.2002E-01

-0.7701E-01 0.4744E-02 0.2165E-02 -0.8538E-02 -0.3539E-01 -0.1632E-02 -0.6954E-03

-0.4663E-02 0.1381E-02 -0.1599E-02 -0.2514E-01 -0.9058E-03 -0.4986E-03 0.5249E-03

0.4369E-01 -0.3276E-02 -0.2731E-03 -0.6431E-03 0.2844E-01 1.032 0.2738E-01

0.2465E-01 -0.4330E-01 0.1349E-02 0.7079E-03 -0.1114E-02 0.8279E-03 -0.1147E-02

0.2654E-01 11.59 0.2370 0.2432 0.1908 1.796 0.4569E-01

0.5911E-01 -0.2752 0.1163E-01 0.6655E-02 0.1226E-01 0.1842E-01 0.1013E-02

0.1045E-02 -0.5404E-02 0.7722E-02 0.8686E-02 0.4204E-01 0.1071E-02 0.1562E-02

-0.4825E-02 0.1291 -0.7895E-01 -0.3190E-02 -0.1356E-02 0.9782E-02 1.391

0.3897E-01 0.4140E-01 -0.1326 0.1479E-02 0.1124E-02 -0.3741E-02 0.1285E-02

-0.3616E-02 0.2953E-01 20.56 0.5560 0.5731 -0.6843 0.7305

0.2406E-01 0.4680E-01 0.3018E-01 0.1988E-01 0.1599E-01 -0.2887E-01 0.2563E-01

-0.4890E-03 0.1693E-02 -0.3787E-02 0.1610E-01 -0.1922E-01 0.2705E-01 0.6995E-03

0.1439E-02 -0.2149E-02 0.6893 -0.2031 -0.8243E-02 -0.7854E-02 0.8168E-01

4.398 0.1241 0.1144 -0.1733 0.4284E-02 0.3585E-02 -0.5150E-02

0.3513E-02 -0.3947E-02 0.1259 12.93 0.3446 0.3637 -0.4216

0.4785E-01 0.1194E-01 0.1851E-01 0.1536E-01 -0.3895E-02 1.157 .

;

%macro Submatrix(initval,dim);

data have;

array SubM(&dim,&dim);

n=&initval;

do i=1 to &dim;

do j=1 to &dim;

if i Le j then do;

SubM(i,j)=n;

n+1;

end;

else SubM(i,j)=SubM(j,i);

end;

do i=1 to &dim;

do j=1 to &dim;

put SubM(i,j) 3.0 @;

end;

put;

end;

run;

%mend Submatrix;

%submatrix(1,8)

%submatrix(37,4)

%submatrix(47,5)

432 data have;

433 input v1-v7;

434 cards;

NOTE: The data set WORK.HAVE has 17 observations and 7 variables.

NOTE: DATA statement used (Total process time):

real time 0.03 seconds

cpu time 0.01 seconds

452 ;

453 %macro Submatrix(initval,dim);

454 data have;

455 array SubM(&dim,&dim);

456 n=&initval;

457 do i=1 to &dim;

458 do j=1 to &dim;

459 if i Le j then do;

460 SubM(i,j)=n;

461 n+1;

462 end;

463 else SubM(i,j)=SubM(j,i);

464 end;

465 end;

466 do i=1 to &dim;

467 do j=1 to &dim;

468 put SubM(i,j) 3.0 @;

469 end;

470 put;

471 end;

472 run;

473 %mend Submatrix;

474

475 %submatrix(1,8)

1 2 3 4 5 6 7 8

2 9 10 11 12 13 14 15

3 10 16 17 18 19 20 21

4 11 17 22 23 24 25 26

5 12 18 23 27 28 29 30

6 13 19 24 28 31 32 33

7 14 20 25 29 32 34 35

8 15 21 26 30 33 35 36

NOTE: The data set WORK.HAVE has 1 observations and 67 variables.

NOTE: DATA statement used (Total process time):

real time 0.03 seconds

cpu time 0.03 seconds

476 %submatrix(37,4)

37 38 39 40

38 41 42 43

39 42 44 45

40 43 45 46

NOTE: The data set WORK.HAVE has 1 observations and 19 variables.

NOTE: DATA statement used (Total process time):

real time 0.03 seconds

cpu time 0.03 seconds

477 %submatrix(47,5)

47 48 49 50 51

48 52 53 54 55

49 53 56 57 58

50 54 57 59 60

51 55 58 60 61

NOTE: The data set WORK.HAVE has 1 observations and 28 variables.

NOTE: DATA statement used (Total process time):

real time 0.03 seconds

cpu time 0.03 seconds

Re: How to extract submatrix?

CTorres — Mon, 03 Nov 2014 16:24:35 GMT

Hi Zana,

Let me first say that I do not understand the "rules" to decide how many submatrixes to get (one?, two?, three?, four?), the diference between Submatrix? and Main_Submatrix? and the dimension of them (4*4?, 8*8?, 5*5?) based on the dimension of the source matrix (8*4 as in the originat request? or 17*7 as in the second request?). I am completly lost about what those numbers mean and what is the purpose of the submatrixes. (????) You did not explain enough.

What I think I understood looking at the sample square submatrices you provided (but maybe I am wrong) is how to fill the them with consecutive integer numbers starting at an initial value. I coded the algorithm in the Submatrix macro.

With that said, and if I understand what you need (I am not sure) I suggest the following approach to solve your problem:

1. Convert the source matrix in a vector: for example the matrix M(8,4) would become a vector V(32) and the matrix M(17*7) a vector V(119).
2. Using the relevant logic in the Submatrix macro obtain the indexes to use to fill the target submatrix
3. Use those indexes to get the number (in the vector) to fill each cell of the submatrix.
4. Output the submatrix in the format you want or need (SAS Table?, report?).

Program to perform Step 1:

data have;
input X1-X7;
cards;
   2.826      0.1117      0.5655E-01 -0.2051     -0.9951     -0.3791E-01 -0.2002E-01
-0.7701E-01 0.4744E-02 0.2165E-02 -0.8538E-02 -0.3539E-01 -0.1632E-02 -0.6954E-03
-0.4663E-02 0.1381E-02 -0.1599E-02 -0.2514E-01 -0.9058E-03 -0.4986E-03 0.5249E-03
0.4369E-01 -0.3276E-02 -0.2731E-03 -0.6431E-03 0.2844E-01   1.032      0.2738E-01
0.2465E-01 -0.4330E-01 0.1349E-02 0.7079E-03 -0.1114E-02 0.8279E-03 -0.1147E-02
0.2654E-01   11.59      0.2370      0.2432      0.1908       1.796      0.4569E-01
0.5911E-01 -0.2752      0.1163E-01 0.6655E-02 0.1226E-01 0.1842E-01 0.1013E-02
0.1045E-02 -0.5404E-02 0.7722E-02 0.8686E-02 0.4204E-01 0.1071E-02 0.1562E-02
-0.4825E-02 0.1291     -0.7895E-01 -0.3190E-02 -0.1356E-02 0.9782E-02   1.391
0.3897E-01 0.4140E-01 -0.1326      0.1479E-02 0.1124E-02 -0.3741E-02 0.1285E-02
-0.3616E-02 0.2953E-01   20.56      0.5560      0.5731     -0.6843      0.7305
0.2406E-01 0.4680E-01 0.3018E-01 0.1988E-01 0.1599E-01 -0.2887E-01 0.2563E-01
-0.4890E-03 0.1693E-02 -0.3787E-02 0.1610E-01 -0.1922E-01 0.2705E-01 0.6995E-03
0.1439E-02 -0.2149E-02 0.6893     -0.2031     -0.8243E-02 -0.7854E-02 0.8168E-01
   4.398      0.1241      0.1144     -0.1733      0.4284E-02 0.3585E-02 -0.5150E-02
0.3513E-02 -0.3947E-02 0.1259       12.93      0.3446      0.3637     -0.4216
0.4785E-01 0.1194E-01 0.1851E-01 0.1536E-01 -0.3895E-02   1.157 .
;
run;
data vector (keep= V:);
Array v(119);
array x(7);
i=0;
do until (eof);
    set have end=eof;
    do j=1 to 7;
       v(i*7+j)=x(j);
    end;
    i=i+1;
end;
output;
stop;
run;

Modified Submatix macro to perform steps 2 to 4 creating three SAS Tables: Subm_1_8, Subm_37_4 and Subm_47_5:

%macro Submatrix(initval,dim,vectdim);
data SUBM_&initval._&dim (keep=var:);
   array var(&dim);
   array v(&vectdim);
   Set vector;
   array SubM(&dim,&dim);
   n=&initval;
   do i=1 to &dim;
      do j=1 to &dim;
         if i Le j then do;
            SubM(i,j)=v(n);
            n+1;
         end;
         else SubM(i,j)=SubM(j,i);
      end;
   end;
   do i=1 to &dim;
      do j=1 to &dim;
         var(j)=SubM(i,j);
      end;
      output;
   end;
run;
%mend Submatrix;

%submatrix(1,8,119)
%submatrix(37,4,119)
%submatrix(47,5,119)

I hope this helps,

CTorres

Re: How to extract submatrix?

Rick_SAS — Mon, 03 Nov 2014 16:35:13 GMT

And please clarify whether you want the solution in SAS/IML or in the DATA step. I believe that I have provided a solution in the SAS/IML language.

Re: How to extract submatrix?

zana — Tue, 04 Nov 2014 17:29:41 GMT

ِِDear Rick, thank you so much. CTorres program was suitable, but as i had said my output file contain more tables (near 4000 table) so i think for solve that i need a loop program. However please let me i have your idea by SAS/IML.

Best regards

Re: How to extract submatrix?

CTorres — Tue, 04 Nov 2014 17:53:24 GMT

I do not understand when you say "my output file contain more tables". It is very difficult (at least for me) to help you because I do not understand the problem. Sorry.

Re: How to extract submatrix?

zana — Wed, 05 Nov 2014 18:52:34 GMT

Dear Torres, so thanks for your kinds.

I'm so sorry for taking your time. I'm sending the output file (contain 3600 different table, all of them had 7 columns*17 rows). Please see it and let me i have your valuable ideas for solve that.

Best regards

Re: How to extract submatrix?

CTorres — Wed, 05 Nov 2014 19:03:13 GMT

Hi Zana,

I could not open the attached file.

Re: How to extract submatrix?

zana — Sun, 09 Nov 2014 12:41:08 GMT

Hi Torres,

You could't open that by TextPad, Notepad or other software? OK, for simplicity this time i sent you as Excel file. But i should inform that in this case (as excel file) you can find 200 row, where each row considered as a table (in other word i changed each table into a row). Regardless the first column (where determine the number of rows), each table (here row) has 118 columns.

Best regards

Zana

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49
50	51	52	53	54	55	56
57	58	59	60	61	62	63
64	65	66	67	68	69	70
71	72	73	74	75	76	77
78	79	80	81	82	83	84
85	86	87	88	89	90	91
92	93	94	95	96	97	98
99	100	101	102	103	104	105
106	107	108	109	110	111	112
113	114	115	116	117	118	.

1	2	3	4	5	6	7	8
2	9	10	11	12	13	14	15
3	10	16	17	18	19	20	21
4	11	17	22	23	24	25	26
5	12	18	23	27	28	29	30
6	13	19	24	28	31	32	33
7	14	20	25	29	32	34	35
8	15	21	26	30	33	35	36