Problems & Puzzles: Puzzles

 Puzzle 992. primes on the faces of a dice Fausto Morales sent the following puzzle   Allocate in each of the 6 faces of a dice, a matrix NxN with 6N^2 digits 0-9 to produce 12N distinct primes one per each row and per each column in all the matrixes. Do this producing the minimal sum of all of these 12N primes.   Note. In each face the primes must be read from left to right in the rows and from top to bottom in the columns.   Q1. Do this for N=3, 4, 5 and 6 Q2. Redo Q1 adding the condition that all the primes bust be emirps.   Example for Q1, N=3, sent by Fausto Morales. No claim that the sum of this example is minimal.   Face 1  6 1 3  4 0 1  1 9 1    Face 2   6 6 1   5 0 3   9 1 9   Face 3  1 3 1  5 4 7  1 9 9   Face 4  1 3 7  4 0 9  9 7 7   Face 5  4 7 9  2 3 3  1 9 7   Face 6  1 1 3  5 2 1  7 7 3

Contributions came from Emmanuel Vantieghem, Michael Hürter and Paul Cleary. All during the week 1-6 March, 2020.

Emmanuel wrote:

The total for  N = 3  is certainly minimal [ n.b. CR. check the Michael's result below] but the other totals are maybe minimal.

Q1

N = 3.  Total  10020
1 1 3,
3 0 7
1 3 9

2 1 1
2 5 7
3 1 3

3 1 1
3 4 7
1 9 9

2 4 1
5 0 3
1 9 7

5 2 1
2 2 9
3 7 3

2 3 3
6 4 1
3 9 7

N = 4   Total  100052

2 1 1 3
1 0 3 9
1 2 0 1
1 1 1 7

1 1 2 3
2 0 2 9
3 3 0 1
1 1 7 1

1 2 1 3
1 0 1 3
5 1 0 1
1 1 9 3

2 2 1 3
1 0 6 1
3 0 0 1
1 3 1 9

2 3 1 1
2 0 3 9
2 8 0 1
1 3 7 3

2 4 2 3
1 0 3 3
4 0 0 1
1 3 9 9

N = 5  Total   1166074
2 1 2 1 1
2 3 2 0 1
1 0 5 3 1
1 0 0 0 7
1 1 1 1 3

1 2 2 1 1
6 0 1 0 1
1 1 1 3 1
1 0 0 3 7
1 1 1 1 7

2 2 5 1 1
5 1 0 0 1
1 0 1 3 3
1 0 0 0 9
1 1 1 1 9

3 1 5 1 1
5 2 2 0 1
1 1 5 0 3
1 0 0 6 1
1 1 1 7 1

4 1 4 1 1
4 6 6 0 1
1 0 3 3 3
1 0 0 3 9
1 1 1 7 3

2 3 3 1 1
6 7 9 0 1
1 2 3 0 1
1 0 0 6 9
1 1 1 7 7

N = 6   Total  15010394
1 1 2 1 1 1
3 2 5 0 0 1
1 1 2 5 0 1
1 0 0 4 1 7
1 0 0 0 5 7
1 1 1 1 1 9

3 2 2 1 1 1
1 0 2 0 0 1
1 2 6 0 0 1
1 0 0 5 1 7
1 0 0 0 0 3
1 1 1 1 9 1

5 4 2 1 1 1
1 0 3 0 0 1
1 3 1 1 0 1
1 0 0 6 7 3
1 0 0 0 4 3
1 1 1 3 1 7

3 5 2 1 1 1
6 0 7 0 0 1
1 2 3 0 0 1
1 0 0 9 2 7
1 0 0 1 0 3
1 1 1 3 7 3

6 1 6 1 1 1
1 5 0 0 0 1
1 3 0 4 1 1
1 0 1 8 3 9
1 0 0 2 7 1
1 1 1 7 7 3

7 4 2 1 1 1
2 1 7 0 0 1
1 2 9 4 0 3
1 0 0 3 6 1
1 0 0 2 9 1
1 1 1 7 9 1

Q1 for N = 7.  Total = 156340024

2 2 2 1 1 1 1
4 1 4 0 0 0 1
1 0 7 4 0 0 1
1 0 0 2 1 0 1
1 0 0 0 0 0 3
1 0 0 0 0 3 9
1 1 1 1 3 3 3

4 1 2 1 1 1 1
2 0 2 0 0 0 1
1 2 3 1 0 0 1
1 0 0 1 5 0 1
1 0 0 0 2 1 3
1 0 0 0 0 3 7
1 1 1 1 3 3 9

3 2 5 1 1 1 1
2 1 1 0 0 0 1
2 3 5 6 0 0 1
1 0 1 2 7 0 3
1 0 0 0 2 1 1
1 0 0 0 0 9 9
1 1 1 1 3 9 9

2 2 5 1 1 1 1
3 3 0 0 0 0 1
4 2 8 2 0 0 1
1 0 2 7 2 1 1
1 0 0 0 5 3 7
1 0 0 0 1 1 7
1 1 1 1 7 1 1

1 3 5 1 1 1 1
1 1 2 0 0 0 1
7 5 6 3 0 0 1
1 0 0 6 3 0 1
1 0 0 0 4 2 9
1 0 0 0 1 8 3
1 1 1 1 7 9 3

2 6 1 1 1 1 1
5 1 3 3 0 0 1
2 0 3 5 0 0 1
1 0 0 0 4 0 3
1 0 0 0 1 5 9
1 0 0 0 1 8 7
1 1 1 1 9 9 1

Q2

For N = 3  and  N = 4  I could not find a decent dice.
Could this be because there are not enough emirps ?

N = 5   Total  1886524
1 1 7 3 1
3 2 4 1 1
7 3 2 7 7
1 0 0 9 1
1 1 1 9 7

7 1 1 7 1
1 1 0 7 1
7 7 2 3 7
1 0 4 5 7
1 1 7 1 7

3 3 3 9 1
3 1 1 2 1
9 6 3 7 7
1 0 0 6 7
1 1 7 1 9

1 1 9 7 1
7 2 3 5 3
9 6 4 4 3
1 1 0 0 3
1 1 7 7 7

7 3 3 3 1
9 0 9 1 1
1 0 0 3 9
1 1 4 2 3
1 1 7 7 9

9 1 7 1 1
3 5 0 8 3
9 1 2 2 9
1 0 4 5 3
1 1 9 3 3

N = 6  Total  26062184
9 1 9 1 1 1
1 3 0 2 0 1
1 5 6 0 1 1
1 1 9 4 1 7
1 0 0 0 4 9
1 1 1 1 1 9

1 3 3 7 1 1
3 2 1 3 0 1
1 5 9 4 2 1
3 0 2 4 5 9
1 0 0 2 6 7
1 1 1 3 3 7

3 3 1 7 1 1
7 0 0 3 0 3
1 1 2 2 1 3
3 6 7 2 7 3
1 0 0 2 7 1
1 1 1 9 1 9

7 3 3 1 1 1
7 2 0 1 0 1
9 3 1 9 0 7
1 1 7 6 4 3
1 0 0 1 8 3
1 1 3 1 3 1

3 7 7 7 1 1
9 1 1 1 0 1
1 2 3 2 1 7
7 3 9 6 0 1
1 0 0 5 2 3
1 1 3 1 7 3

7 9 7 7 1 1
9 2 3 2 0 1
7 1 1 4 2 7
3 0 8 1 3 7
1 0 0 1 2 9
1 1 3 7 9 7

N = 7.  Total = 277900958

9 3 3 1 1 1 1
1 0 3 9 0 0 1
7 0 4 1 1 0 1
3 1 8 2 6 5 3
1 0 0 3 0 9 7
1 0 0 0 2 1 1
1 1 1 1 3 3 9

3 7 7 3 1 1 1
7 1 1 0 0 0 1
3 0 4 2 2 0 3
3 2 3 7 8 5 1
1 0 0 0 2 9 1
1 0 0 0 0 3 3
1 1 1 1 7 1 1

9 7 3 7 1 1 1
7 0 0 2 1 0 1
1 1 3 1 7 0 1
9 1 6 2 3 4 3
1 0 0 7 8 5 7
1 0 0 0 0 3 7
1 1 1 3 3 7 3

1 3 1 1 3 1 1
1 1 2 0 0 0 1
7 5 3 9 0 0 1
1 6 1 6 1 1 9
1 0 0 3 6 0 1
1 0 0 0 1 1 7
1 1 1 7 3 7 9

3 3 3 3 3 1 1
7 0 5 3 0 0 1
9 3 7 0 6 0 1
7 1 3 2 2 6 7
1 0 0 2 2 9 9
1 0 0 0 0 3 9
1 1 1 7 9 7 3

3 1 3 7 3 1 1
9 3 2 1 1 0 1
9 2 3 5 0 0 3
7 6 1 0 5 0 1
1 0 0 0 8 4 9
1 0 0 0 7 2 1
1 1 1 9 7 9 9

***

Michael wrote:

Q1. I found the following result for n = 3:

Sum = 9400

face 1:
113
307
139

face 2:
223
163
137

face 3:
311
109
317

face 4:
241
349
331

face 5:
431
257
193

face 6:
127
521
179

(N.B. Carlos Rivera checked the validity of the Michael's solution: 36 distinct primes, all are 3 digits type, the total sum is 9400. Please notice that Michael does not claim that 9400 is the minimal sum possible).

***

Paul wrote:

Q1.  N=3.

I think this is the minimum sum;

Grand Total 9882 ,

Face  1,   113

307

139

Total 1172

Face 2  211

257

313

Total 1328

Face 3 311

503

317

Total 1722

Face 4  241

349

331

Total 1788

Face 5  421

337

199

Total 1806

Face 6  521

229

373

Total 2066

There is a second solution for face 1 that gives the same total

Face 1  131

103

379

***

On March 7, 2020 Michael wrote:

I have computed the optimal solution for n = 3. I have checked the result.

Sum = 9382

face 1
311
103
317... 1282

face 2
211
229
379... 1368

face 3
113
503
191... 1398

face 4
163
307
173... 1754

face 5
421
353
179... 1780

face 6
233
443
197... 1800

***

On March 13, Vladimir Shirokov wrote:

I think these are the minimum sums for n=3 and n=4

Q1.3 - there are two solutions with 9382 sum

 Dice 1 Dice 2 151 109 337       :1410   311 103 317       :1282 211 229 379       :1368 131 607 373       :1754 421 353 179       :1780 241 349 331       :1788 151 103 337       :1344 311 109 317       :1348 211 229 379       :1368 131 607 373       :1754 421 353 179       :1780 241 349 331       :1788

NB by CR: Please notice that the two solutions by Shirokov are distinct to the one gotten above by Hürter. So we have 3 distinct minimal solutions.

Q1.4 one solution with sum 92126

2111
1103
1097
3313
:13304

1321
1013
2141
3119
:15188

1231
2003
1049
3319
:15258

2131
3001
1087
1931
:15724

2221
2011
1151
3137
:16014

4211
1021
1129
1733
:16638

New question: Can you get a lower sum solutions for Q1. n=4?

***

Michael Hürter wrote on March 19, 2020:

For n = 3 I found 192 distinct optimal solutions with sum 9382. See this file.

***

Gennady Gusev wrote on March 20, 2020:

I am very sorry, but there are mistakes in V. Shirokov's solutions with sum 92126.

Face 1
2111
1103
1097
3313
:13304
contains 2 identical numbers: 1103 in the 2nd row and in the 2nd column.

Face 2
1321
1013
2141
3119
:15188

the number 2141 in the 3rd row & in the 3rd column.
I found 2 the smallest solutions to Q1.4 with sum  92706.

2111  1321  2221  1123  2213  4211
1103  2011  1013  2203  3001  1021
3037  1151  1061  3023  1511  1129
1913  3137  3319  1171  1399  1733
-----------  ----------  -----------  ---------  ----------  ---------
13808 15114 15210 15310 16626 16638

2213  1321  2311  1123  2131  3221
1013  2011  1021  2141  2003  1201
1151  1051  1187  3041  2039  2029
1733  3137  3191  1319  1979  1373
----------- ----------   ----------- ---------   ----------   ---------
14704 15004 15294 15518 15818 16368

***

Gennady Gusev wrote on March 27, 2020:

I found several (more) solutions for N=4 with sum=92706.
I think they are optimal.

2111 1213 2113 1231 2213 4211

1103 3011 2003 1201 3001 1021

3037 2153 2161 2027 1511 1129

1913 1117 1319 3331 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1213 2113 1123 2213 4211

1103 3011 2003 2203 3001 1021

3037 2153 2161 3023 1511 1129

1913 1117 1319 1171 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1213 2221 1231 2213 4211

1103 3011 1013 1201 3001 1021

3037 2153 1061 2027 1511 1129

1913 1117 3319 3331 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1213 2221 1123 2213 4211

1103 3011 1013 2203 3001 1021

3037 2153 1061 3023 1511 1129

1913 1117 3319 1171 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1321 2113 1231 2213 4211

1103 2011 2003 1201 3001 1021

3037 1151 2161 2027 1511 1129

1913 3137 1319 3331 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1321 2113 1123 2213 4211

1103 2011 2003 2203 3001 1021

3037 1151 2161 3023 1511 1129

1913 3137 1319 1171 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1321 2221 1231 2213 4211

1103 2011 1013 1201 3001 1021

3037 1151 1061 2027 1511 1129

1913 3137 3319 3331 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2111 1321 2221 1123 2213 4211

1103 2011 1013 2203 3001 1021

3037 1151 1061 3023 1511 1129

1913 3137 3319 1171 1399 1733

------- -------- -------- ------- ------ --------

13808 15114 15210 15310 16626 16638

2213 1321 2311 1123 2131 3221

1013 2011 1021 2141 2003 1201

1151 1051 1187 3041 2039 2029

1733 3137 3191 1319 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2311 1123 2131 3121

1013 2011 1021 2141 2003 2203

1151 1051 1187 3041 2039 2027

1733 3137 3191 1319 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2311 1231 2131 3221

1013 2011 1021 1103 2003 1201

1151 1051 1187 2441 2039 2029

1733 3137 3191 3119 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2311 1231 2131 3121

1013 2011 1021 1103 2003 2203

1151 1051 1187 2441 2039 2027

1733 3137 3191 3119 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2113 1123 2131 3221

1013 2011 3011 2141 2003 1201

1151 1051 1289 3041 2039 2029

1733 3137 1171 1319 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2113 1123 2131 3121

1013 2011 3011 2141 2003 2203

1151 1051 1289 3041 2039 2027

1733 3137 1171 1319 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2113 1231 2131 3221

1013 2011 3011 1103 2003 1201

1151 1051 1289 2441 2039 2029

1733 3137 1171 3119 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1321 2113 1231 2131 3121

1013 2011 3011 1103 2003 2203

1151 1051 1289 2441 2039 2027

1733 3137 1171 3119 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2311 1123 2131 3221

1013 3001 1021 2141 2003 1201

1151 2153 1187 3041 2039 2029

1733 1117 3191 1319 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2311 1123 2131 3121

1013 3001 1021 2141 2003 2203

1151 2153 1187 3041 2039 2027

1733 1117 3191 1319 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2311 1231 2131 3221

1013 3001 1021 1103 2003 1201

1151 2153 1187 2441 2039 2029

1733 1117 3191 3119 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2311 1231 2131 3121

1013 3001 1021 1103 2003 2203

1151 2153 1187 2441 2039 2027

1733 1117 3191 3119 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2113 1123 2131 3221

1013 3001 3011 2141 2003 1201

1151 2153 1289 3041 2039 2029

1733 1117 1171 1319 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2113 1123 2131 3121

1013 3001 3011 2141 2003 2203

1151 2153 1289 3041 2039 2027

1733 1117 1171 1319 1979 1193

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2113 1231 2131 3221

1013 3001 3011 1103 2003 1201

1151 2153 1289 2441 2039 2029

1733 1117 1171 3119 1979 1373

------- -------- -------- ------- ------ --------

14704 15004 15294 15518 15818 16368

2213 1213 2113 1231 2131 3121

1013 3001 3011 1103 2003 2203

1151 2153 1289 2441 2039 2027

1733 1117 1171 3119 1979 1193

-------  --------  --------  -------  ------  --------

14704 15004 15294 15518 15818 16368

And

I found several solutions for N=5 with sum=957734.
I think they are optimal.

957734
21211 11113 11131 21121 11321 12113
21001 10301 20011 12011 30011 10303
10223 31013 21101 21011 12203 21013
10061 11059 10337 20047 21139 10091
13177 11117 13337 11731 11119 31337
------  ------ ------ -------  --------  --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 11131 21121 11321 11213
21001 10301 20011 12011 30011 20101
10223 31013 21101 21011 12203 13003
10061 11059 10337 20047 21139 10193
13177 11117 13337 11731 11119 33317
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 11131 21121 13121 12113
21001 10301 20011 12011 10211 10303
10223 31013 21101 21011 30211 21013
10061 11059 10337 20047 21031 10091
13177 11117 13337 11731 11399 31337
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 11131 21121 13121 11213
21001 10301 20011 12011 10211 20101
10223 31013 21101 21011 30211 13003
10061 11059 10337 20047 21031 10193
13177 11117 13337 11731 11399 33317
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 12211 21121 11321 12113
21001 10301 10103 12011 30011 10303
10223 31013 10133 21011 12203 21013
10061 11059 31033 20047 21139 10091
13177 11117 11177 11731 11119 31337
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 12211 21121 11321 11213
21001 10301 10103 12011 30011 20101
10223 31013 10133 21011 12203 13003
10061 11059 31033 20047 21139 10193
13177 11117 11177 11731 11119 33317
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 12211 21121 13121 12113
21001 10301 10103 12011 10211 10303
10223 31013 10133 21011 30211 21013
10061 11059 31033 20047 21031 10091
13177 11117 11177 11731 11399 31337
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

957734
21211 11113 12211 21121 13121 11213
21001 10301 10103 12011 10211 20101
10223 31013 10133 21011 30211 13003
10061 11059 31033 20047 21031 10193
13177 11117 11177 11731 11399 33317
------ ------ ------ ------- -------- --------
150572 150574 150574 161564 171766 172684

***

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