IQ Link is a puzzle produced by Smart Games. The set includes 12 pieces that, after some effort, can be all made to fit on the board. The bundled booklet offers 120 challenges (partially filled boards that have to be completed).

Although the booklet provides solutions to all puzzles, it only left me with more questions. How many solutions are there in total? Are all solutions fully linked? Are there any single piece placements with unique solution?

One day I decided to try enumerating the solutions. Fortunately the search space turned out to be reasonably small. On this page I summarize my findings and provide random puzzle selector.

**Note about linkedness.**
Quote from the boolket: *"Hint: in the solution all puzzle pieces will be linked to each other,
resulting in 1 group of 12 connected puzzle pieces."*
This condition (probably related to the name "IQ Link") seemed too strict to me,
so I also explored the relaxed variant of the puzzle, where solutions don't have to be fully linked.

**Note about symmetry.**
Since the board has 2-way rotational symmetry,
each placement of pieces can be rotated 180° while remaining essentially the same.
Therefore, when counting piece placements,
we can either ignore symmetry (count all placements separately from each other) to obtain *total number of placements*.
Or we can use symmetry (count pairs of mutually symmetrical placements) to get *number of unique placements*
(twice smaller than the total).

Choose rule variant and number of pieces, to get a random puzzle:

Strict (only fully-linked solutions): | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Relaxed (any solutions): | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

**Any solutions.**
There are **50,000** solutions to the relaxed version of the puzzle
(counting all solutions, whether linked or not), or **25,000** unique solutions.
It's interesting that this round number emerges from the arbitrary-looking ruleset.

**Linked solutions.**
When considering only fully-linked solutions,
there are **21,952** solutions in total, or **10,976** unique solutions.
This number, while less round, is not any less remarkable:
25,000 = 2^{3}×5^{5}, and similarly,
10,976 = 2^{5}×7^{3}.
In other words, both have very compact factoring.
If anyone has good explanation for this, I'd be very curious to hear it.

A puzzle is a partially filled board, that allows only unique solution. Under this definition the puzzles can be counted, separately for two rule variants:

Pieces | Number of puzzles Strict (fully-linked solutions) | Number of puzzles Relaxed (any solutions) |
---|---|---|

1 | 20 | 18 |

2 | 57,032 | 64,624 |

3 | 792,304 | 1,243,710 |

4 | 3,199,086 | 5,732,294 |

5 | 7,100,096 | 13,672,638 |

6 | 10,335,336 | 20,839,986 |

7 | 10,477,190 | 21,821,930 |

8 | 7,500,464 | 16,009,210 |

9 | 3,735,984 | 8,132,178 |

10 | 1,235,874 | 2,735,178 |

11 | 244,604 | 549,426 |

total | 44,677,990 | 90,801,192 |

These are the total numbers (ignoring symmetry). To get the numbers of unique puzzles of each type, divide each number by 2.

(These questions are explored in the relaxed rule variant).

**Are there any single-piece puzzles?**
In other words, are there any single piece placements that have unique solution?
- There are 18 such puzzles
(redundant now, since these puzzles are included in the selector above).

**What is the most frequent placement of a single piece?** - Here.
**Of two pieces?** - Here.

**Are there solutions with two empty points?**
- Yes, 13 pairs.

**Are there any 3-cluster solutions?**
- Yes, 3,064 such solutions (1,532 pairs) exist.
All other 24,984 non-fully linked solutions (12,492 pairs) have 2 clusters.

Among the 3-cluster solutions, 30 (15 pairs) don't have any single-piece clusters.