TomasSirgedas

references:
https://www.facebook.com/groups/391950357895182/permalink/1748946985528839/
https://www.puzzlefun.online/problems
https://erich-friedman.github.io/mathmagic/0802.html

For each N, try to pack 1 1x1 square, 2 2x2 squares, 3 3x3 squares, 4 4x4 squares, ..., N NxN squares into a N*(N+1)/2xN*(N+1)/2 square

The goal is to omit as little area as possible. (e.g. omitting 1 2x2 and 2 3x3s means that a total area of 4+9+9 is omitted)

Here are the optimal results for small N:

TomasSirgedas

FILN;PUXY;TVWZ
.000002.
22212221
20111021
20010011
00220221
11120122
10122112
.000011.

TomasSirgedas

FIL;NPVWX;TUYZ
00000112  
11121122  
12221002  
120..012  
000..011  
10221021  
11211121  
11221222  

TomasSirgedas

basis = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 }

46 PTs with legs < 10^9 are listed below.

        {0,1,0,1,1,1,1,1,0,1,0,0} -- this coloring scheme avoids mono-coloring all 46 PTs
                                     (parity of [dot product of this with vector-form of number] determines the number's color)

---
     12 {2,1,0,0,0,0,0,0,0,0,0,0}
     35 {0,0,1,1,0,0,0,0,0,0,0,0}

TomasSirgedas

primitive pythagorean triples using only factors of { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 }
and legs < 10^9

12 35 37
12 5 13
132 475 493
84 187 205
84 13 85
156 133 205
204 253 325

TomasSirgedas

01 {33,56,65}                               * {15,8,17}                                + 1073                {264,448,495,520,561,840,952,975,1105,1073}
11 {13,84,85}                               * {9,40,41}                                + 925                 {117,520,533,756,765,3360,3400,3444,3485,925}
01 {153,104,185}                            * {13,84,85}                               + 9061                {1352,1989,2405,8736,8840,12852,13005,15540,15725,9061}
10 {161,240,289}                            * {187,84,205}                             + 38675               {13524,20160,24276,30107,33005,44880,49200,54043,59245,38675}
01 {297,304,425}                            * {105,208,233}                            + 77393               {31185,31920,44625,61776,63232,69201,70832,88400,99025,77393}
01 {153,104,185}                            * {105,88,137}                             + 21473               {9152,10920,13464,14248,16065,16280,19425,20961,25345,21473}
00 {185,672,697}                       

TomasSirgedas

reference: https://dustingmixon.wordpress.com/2021/02/01/polymath16-seventeenth-thread-declaring-victory/#comment-46997

mono-edge densities for n: (tentative)
3 - 0.333333 (or no solution)
4 - 0.416667
5 - 0.538462
6 - 0.571429
7 - 0.633333
8 - 0.666667
9 - 0.701149 -- I only tried like 5 shapes

TomasSirgedas

On the A2 lattice (triangular lattice -- each vertex has 6 neighbors), color the vertices
such that the density of mono edges is maximized. An edge is "mono" if its neighbors are
the same color.

A "forbidden shape" is also provided. No set of vertices in this shape can be all-the-same-color.

https://dustingmixon.wordpress.com/2021/02/01/polymath16-seventeenth-thread-declaring-victory/#comment-46576


O O . . . . .

TomasSirgedas

class PrimeSieve
{
public:
   PrimeSieve( int64_t n )
   {
      m.resize( n, true );
      m[0] = m[1] = false;
      for ( int64_t i = 2; i * i < n; i++ ) if ( isPrime( i ) )
         for ( int64_t j = i*2; j < n; j += i )
            m[j] = false;

TomasSirgedas

FLVX;IPYZ;NTUW
.001122.
00211122
10200002
12202221
12011211
10001201
12021101
.222000.