Tsukuda's Square, also known as 4x4 or "It", is a very difficult type of sliding piece puzzle. It consists of 16 tiles in a 4x4 square. On the left of each row is a plunger, which pushes that row one tile to the right. At the top is a large plunger which pushes the three rightmost columns down one tile. Springs push back the plungers as soon as they are released. If the top plunger is pushed in, the first row plunger can no longer move.
There are several different designs (see also Cubic Circular Issue 2 p9). The version that was marketed in Britain was called "It", and it had tiles that were numbered 1 to 16, and the tiles were also made of four different colours plastic, four each of the colours green, red, yellow, and blue. In the USA the puzzle was marketed by Gabriel under the name "Tsukuda's Square", and it apparently came in several different versions. The most difficult version has black tiles numbered from 1 to 16. A version in which the tiles are split diagonally halfred and halfwhite is shown above as well. There are 4 sets of 4 identical tiles in that version. That design was also done in blue and white. I believe the third version simply has four tiles of four colours.
Tsukuda's Square is closely related to Rubik's Fifteen, but much more difficult because it has fewer plungers. It was invented by Toshihisa Suga, patented on 26 January 1983, GB 2,101,897.
If your browser supports JavaScript, then you can play the 'It' puzzle by clicking the link below:
I have calculated God's Algorithm to find out how many moves each position needs to get solved. This is for the version with diagonally split tiles. A move in this context is any of the 12 ways of cycling seven tiles, as explained in the solution. The solved position has a pattern of concentric diamonds.
Depth  Positions 

0  1 
1  12 
2  93 
3  636 
4  4,174 
5  26,808 
6  165,092 
7  917,693 
8  4,071,834 
9  12,552,545 
10  23,737,340 
11  18,619,383 
12  2,938,388 
13  29,001 
Total  63,063,000 
1a. Cycle all but top left tile clockwise:  TBtb  
1b. Cycle all but top left tile anticlockwise:  BTbt  
2a. Cycle all but top right tile clockwise:  ATBtba  
2b. Cycle all but top right tile anticlockwise:  ABTbta 
These sequences can also be performed on any other pair of adjacent rows.
Phase 1: Solve the first tile of the top row.
Note that the method used in bc above to bring up a tile to the desired row will be used whenever necessary in later phases without further explanation.
Phase 2: Solve the rest of the top row.
In this phase we will use 1a to successively place tile 4, 3 and 2 at the
second position of the first row.
Phase 3: Solve the second row.
This is done using exactly the same method as the first row.
Phase 4: Solve tiles 15 and 16.
Phase 5: Solve tile 13.
Depending on its position, do one of the following:
Row 3, column 1: Do 1a, 2b.
Row 3, column 2: Do 1b three times, 2a twice.
Row 3, column 3: Do 2b twice, 1a three times, 2b once.
Row 3, column 4: Do 2a twice, 1a twice, 2a twice.
Row 4, column 1: Do nothing.
Row 4, column 2: Do 1a twice, 2b twice.
The final phase below uses three new sequences on the bottom two rows:
3a. 
Do 1b three times, 2a twice, 1a twice, 2b twice:
[Swaps pieces (3,1)(3,2) and (3,3)(4,2).] 
DTdt DTdt DTdt C TDtd TDtd c TDtd TDtd C DTdt DTdt c 

3b. 
Do 1b three times, 2b three times, 1b once, 2b once:
[Swaps pieces (3,1)(3,3) and (3,2)(4,2).] 
DTdt DTdt DTdt C DTdt DTdt DTdt c DTdt C DTdt c 

3c. 
Do 1a once, 2a once, 1b twice:
[Swaps pieces (3,2)(3,3) and (3,1)(4,2).] 
TDtd C TDtd c DTdt DTdt 
These sequences do not move the tile at row 3 column 4, and make two swaps amongst the remaining four unsolved tiles.
Phase 6: Solve the remaining tiles.
It is always possible to get one of the positions in the table below (or
if you're lucky even to solve it) by doing one of the sequences 3ac. Then
do the sequence shown in the table.
If this places tile 12 correctly, then you can now solve it by applying
one of 3ac.
If tile 12 was correctly placed before, then the sequence from the table
will have moved it out, and you will have to repeat this phase once more
to solve the puzzle.
14 10 11 9 11 9 14 10 10 14 9 11 9 11 10 14 12 12 12 12 14 10 9 12 14 9 11 12 11 10 
Perform TCtc, 3a, CTct.  
10 12 11 9 9 12 14 10 14 12 9 11 11 12 10 14 14 11 10 9 10 11 9 12 10 14 11 12 9 14 10 12 9 11 14 12 14 9 11 10 
Perform TCtc, 3b, CTct.  
11 10 12 9 14 9 12 10 9 14 12 11 10 11 12 14 14 11 10 9 11 9 10 12 11 10 14 12 14 9 
Perform TCtc, 3c, CTct. 