Fibo puzzles

_{Published: May 29, 2026}

Fibo puzzles are a class of puzzles in which, every time the structure is extended to include one or more boxes, the number of optimal pushes grows exponentially. Asymptotically, as the number of boxes approaches infinity, this expansion occurs at the exact rate of the classic Fibonacci sequence: the golden ratio, or phi (approx. 1.618). This results in unusually long solution lengths. For example, a puzzle of this type with 16 boxes can require more than 20,000 moves and over 8,000 pushes.

The recurrence relation for optimal pushes across distinct versions of this puzzle type (ranging from fewer to more boxes) may resemble the Fibonacci recurrence P(n) = P(n-1) + P(n-2), but that is not always the case.

Some variants require two increments of one box (resulting in two boxes added) because a single increment either makes the puzzle unsolvable or is structurally impossible, requiring a paired addition. In these specific variants, the growth rate measured over two increments is phi + 1. Since phi + 1 = phi^2, we can calculate a theoretical growth rate per single increment using sqrt(phi^2) = phi. Thus, it remains the golden ratio.

Picokosmos 12. Created in October 2001 by Aymeric du Peloux and titled Hanoï in French. It resembles the Tower of Hanoi and contains 7 boxes.

• Recurrence formula: P(n) = P(n-1) + P(n-2) + 6 - (-1)^n for n >= 6
• Base cases: P(4) given as 23, P(5) given as 43

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Picokosmos 17. Created in December 2001 by Aymeric du Peloux and titled Grand Hanoï in French. It contains 8 boxes, but it is not a version of Picokosmos 12 expanded to 8 boxes; Picokosmos 17 is more compact.

• Recurrence formula: P(n) = 3 * P(n-2) - P(n-4) + 11 for n >= 8
• Base cases: P(4) given as 15, P(6) given as 57

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Femtocosmos 3. Created in February 2026 by Aymeric du Peloux, it introduces a novel structure shaped like a 'T' rather than the traditional 'I' form.

• Recurrence formula: P(n) = 3 * P(n-2) - P(n-4) + 2n - 5 for n >= 13
• Base cases: P(9) given as 75, P(11) given as 221

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Picokosmos 17 (David Holland variant). It appeared in a 2002 analysis.

• Recurrence formula: P(n) = P(n-1) + P(n-2) + 7 for n >= 6
• Base cases: P(4) given as 19, P(5) given as 35

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Picokosmos 17 (Yang Chao variant). It appeared in a 2012 analysis in Chinese.

• Recurrence formula: P(n) = P(n-1) + P(n-2) + 8 for n >= 6
• Base cases: P(4) given as 18, P(5) given as 34

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Scorpius: In the style of Picokosmos 17. It contains 12 boxes. Created by David Dahlem in 2014, it appears in a collection called DD-1.

• Recurrence formula: P(n) = 3 * P(n-2) - P(n-4) + 4 for n >= 12
• Base cases: P(8) given as 106, P(10) given as 284

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Eiffel Tower: Created by 20603 (Zou Yongzhong). It contains 16 boxes. This version is from a 2016 Chinese forum message.

• Recurrence formula: P(n) = P(n-1) + P(n-2) + n – 1 + 2 * (-1)^(n-1) for n >= 6
• Base cases: P(4) given as 8, P(5) given as 21

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Around 2009, Dries De Clercq created a set called Fibo, featuring 37 variants of Picokosmos 17. I analyzed one of them:

Puzzle 36-9-286b. It contains 9 boxes.

• Recurrence formula: P(n) = 3 * P(n-2) - P(n-4) + n - 15 for n >= 11
• Base cases: P(7) given as 114, P(9) given as 286

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Conclusion

Fibo puzzles can be difficult for both humans and automated solvers because of their large solution lengths. Automated solvers fail to find solutions when the number of boxes is relatively small, for example, above 18.

In some solutions, I observed that the previous solution is fully contained at the end; when optimized, it sometimes shows slight variations. Also, the ending player position is relatively the same across all versions of a puzzle. This presents an opportunity to develop a method that identifies the pattern from the solution of a version of the puzzle with fewer boxes, and repeats it enough times to create a working solution.

The complexity of Fibo puzzles increases every time boxes are added, due to the growing number of pushes required, but if a pattern of repetition were found, the difficulty would theoretically remain the same for all versions. In practice, when treating them as general puzzles, every expansion makes the puzzle significantly harder and much longer to solve.

Update 12 June 2026

I created a script that generates solutions for Picokosmos 17. This is super interesting because every new solution contains the previous one at the end. This shows that it might be possible to do the same for all solutions in this class of puzzles, if a proper pattern is found and followed.

Update 19 June 2026

In the case of the Picokosmos 17 series of variants, I noticed that the previous puzzle version's solution is fully included at the end of the solution if you ignore the case of the letters. In LURD notation, uppercase letters indicate a push. For example, if we replace the last 170 moves of the Picokosmos 17 solution with the corresponding 170 moves of the 6-box variant, the only differences are two specific case shifts: the 'DD' in the 6-box variant becomes 'dD' in Picokosmos 17, and an 'r' becomes an 'R'. This is because what is a push in the 6-box variant becomes a move in the same direction in Picokosmos 17, and vice versa.

At a certain point in Picokosmos 17, the box setup is almost identical to an early state of the previous puzzle, but the subsequent flow of moves, without considering whether a move is a push or a walk, matches perfectly. This explains why the previous puzzle version's solution, regarding steps, is fully incorporated at the end of the solution.

Picokosmos 17 (6-box variant) state after 2 moves:

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Picokosmos 17 state after 326 moves:

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In the case of the Femtocosmos 3 series of variants, I noticed that the previous puzzle version's solution is consistently included at the end of the solution, excluding only the first 9 moves: 'UdldlluRU', which are replaced by 'DluuurDuu'. This sets up a state virtually identical to the previous puzzle's initial progress. Example:

Femtocosmos 2 state after 9 moves:

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Femtocosmos 3 state after 1323 moves:

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Thus, this opportunity for pattern-based solution generation also applies to Femtocosmos 3.

—Analysis and article by Carlos Montiers Aguilera.