A tiling of a square using pieces which are all also squares, all of different sizes.

This was believed impossible for many years until somebody found one. Now it is known that the one with the fewest squares uses 21 squares, and there are hundreds using 26 squares or less.

The lowest order of the squared squares with 21 internal squares is called the pluperfect square. It was discovered in 1978 by a computer scientist and mathemetician named A.J.W. Duijvestijn.

The Bouwkamp code for the pluperfect square is {50, 35, 27}, {8, 19}, {15, 17, 11}, {6, 24}, {29, 25, 9, 2}, {7, 18}, {16}, {42}, {4, 37}, {33}.

The abovementioned pluperfect square {50, 35, 27}, {8, 19}, {15, 17, 11}, {6, 24}, {29, 25, 9, 2}, {7, 18}, {16}, {42}, {4, 37}, {33} is displayed graphically in this ASCII art image. This figure is a square of side 112 divided into smaller squares of side length 2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42 and 50.

(Note that since the characters forming the illustration are not square, I have drawn in one diagonal of each square to prove the figure has been decomposed into squares not rectangles.)



            1         2         3         4         5         6         7         8         9         0         1
  0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901
 0.------------------------------------------------..---------------------------------..-------------------------.
 1|\                                               ||\                                ||\                        |
 2| \                                              || \                               || \                       |
 3|  \                                             ||  \                              ||  \                      |
 4|   \                                            ||   \                             ||   \                     |
 5|    \                                           ||    \                            ||    \                    |
 6|     \                                          ||     \                           ||     \                   |
 7|      \                                         ||      \                          ||      \                  |
 8|       \                                        ||       \                         ||       \                 |
 9|        \                                       ||        \                        ||        \                |
10|         \                                      ||         \                       ||         \               |
 1|          \                                     ||          \                      ||          \              |
 2|           \                                    ||           \                     ||           \             |
 3|            \                                   ||            \                    ||            \            |
 4|             \                                  ||             \                   ||             \           |
 5|              \                                 ||              \                  ||              \          |
 6|               \                                ||               \                 ||               \         |
 7|                \                               ||                \                ||                \        |
 8|                 \                              ||                 \               ||                 \       |
 9|                  \                             ||                  \              ||                  \      |
20|                   \                            ||                   \             ||                   \     |
 1|                    \                           ||                    \            ||                    \    |
 2|                     \                          ||                     \           ||                     \   |
 3|                      \                         ||                      \          ||                      \  |
 4|                       \                        ||                       \         ||                       \ |
 5|                        \                       ||                        \        ||                        \|
 6|                         \                      ||                         \       |'-------------------------'
 7|                          \                     ||                          \      |.------..-----------------.
 8|                           \                    ||                           \     ||\     ||\                |
 9|                            \                   ||                            \    || \    || \               |
30|                             \                  ||                             \   ||  \   ||  \              |
 1|                              \                 ||                              \  ||   \  ||   \             |
 2|                               \                ||                               \ ||    \ ||    \            |
 3|                                \               ||                                \||     \||     \           |
 4|                                 \              |'---------------------------------''------'|      \          |
 5|                                  \             |.-------------..---------------..---------.|       \         |
 6|                                   \            ||\            ||\              ||\        ||        \        |
 7|                                    \           || \           || \             || \       ||         \       |
 8|                                     \          ||  \          ||  \            ||  \      ||          \      |
 9|                                      \         ||   \         ||   \           ||   \     ||           \     |
40|                                       \        ||    \        ||    \          ||    \    ||            \    |
 1|                                        \       ||     \       ||     \         ||     \   ||             \   |
 2|                                         \      ||      \      ||      \        ||      \  ||              \  |
 3|                                          \     ||       \     ||       \       ||       \ ||               \ |
 4|                                           \    ||        \    ||        \      ||        \||                \|
 5|                                            \   ||         \   ||         \     |'---------''-----------------'
 6|                                             \  ||          \  ||          \    |.----..----------------------.
 7|                                              \ ||           \ ||           \   ||\   ||\                     |
 8|                                               \||            \||            \  || \  || \                    |
 9'------------------------------------------------''-------------'|             \ ||  \ ||  \                   |
50.---------------------------..-----------------------..--------..|              \||   \||   \                  |
 1|\                          ||\                      ||\      |'''---------------''----'|    \                 |
 2| \                         || \                     || \     |.-----..----------------.|     \                |
 3|  \                        ||  \                    ||  \    ||\    ||\               ||      \               |
 4|   \                       ||   \                   ||   \   || \   || \              ||       \              |
 5|    \                      ||    \                  ||    \  ||  \  ||  \             ||        \             |
 6|     \                     ||     \                 ||     \ ||   \ ||   \            ||         \            |
 7|      \                    ||      \                ||      \||    \||    \           ||          \           |
 8|       \                   ||       \               |'-------''-----'|     \          ||           \          |
 9|        \                  ||        \              |.--------------.|      \         ||            \         |
60|         \                 ||         \             ||\             ||       \        ||             \        |
 1|          \                ||          \            || \            ||        \       ||              \       |
 2|           \               ||           \           ||  \           ||         \      ||               \      |
 3|            \              ||            \          ||   \          ||          \     ||                \     |
 4|             \             ||             \         ||    \         ||           \    ||                 \    |
 5|              \            ||              \        ||     \        ||            \   ||                  \   |
 6|               \           ||               \       ||      \       ||             \  ||                   \  |
 7|                \          ||                \      ||       \      ||              \ ||                    \ |
 8|                 \         ||                 \     ||        \     ||               \||                     \|
 9|                  \        ||                  \    ||         \    |'----------------''----------------------'
70|                   \       ||                   \   ||          \   |.----------------------------------------.
 1|                    \      ||                    \  ||           \  ||\                                       |
 2|                     \     ||                     \ ||            \ || \                                      |
 3|                      \    ||                      \||             \||  \                                     |
 4|                       \   |'-----------------------''--------------'|   \                                    |
 5|                        \  |.--..-----------------------------------.|    \                                   |
 6|                         \ ||\ ||\                                  ||     \                                  |
 7|                          \|| \|| \                                 ||      \                                 |
 8'---------------------------''--'|  \                                ||       \                                |
 9.-------------------------------.|   \                               ||        \                               |
80|\                              ||    \                              ||         \                              |
 1| \                             ||     \                             ||          \                             |
 2|  \                            ||      \                            ||           \                            |
 3|   \                           ||       \                           ||            \                           |
 4|    \                          ||        \                          ||             \                          |
 5|     \                         ||         \                         ||              \                         |
 6|      \                        ||          \                        ||               \                        |
 7|       \                       ||           \                       ||                \                       |
 8|        \                      ||            \                      ||                 \                      |
 9|         \                     ||             \                     ||                  \                     |
90|          \                    ||              \                    ||                   \                    |
 1|           \                   ||               \                   ||                    \                   |
 2|            \                  ||                \                  ||                     \                  |
 3|             \                 ||                 \                 ||                      \                 |
 4|              \                ||                  \                ||                       \                |
 5|               \               ||                   \               ||                        \               |
 6|                \              ||                    \              ||                         \              |
 7|                 \             ||                     \             ||                          \             |
 8|                  \            ||                      \            ||                           \            |
 9|                   \           ||                       \           ||                            \           |
00|                    \          ||                        \          ||                             \          |
 1|                     \         ||                         \         ||                              \         |
 2|                      \        ||                          \        ||                               \        |
 3|                       \       ||                           \       ||                                \       |
 4|                        \      ||                            \      ||                                 \      |
 5|                         \     ||                             \     ||                                  \     |
 6|                          \    ||                              \    ||                                   \    |
 7|                           \   ||                               \   ||                                    \   |
 8|                            \  ||                                \  ||                                     \  |
 9|                             \ ||                                 \ ||                                      \ |
10|                              \||                                  \||                                       \|
 1'-------------------------------''-----------------------------------''----------------------------------------'

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