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Quizzes > Mathematics > Advanced Math

How Well Do You Know Fibonacci Nim? Take the Quiz!

Ready to conquer the Fibonacci Nim game? Challenge yourself now

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper cut illustration of layered nim sticks in fibonacci sequence on sky blue background for strategy quiz

Think you've got the perfect move? Dive into our Fibonacci Nim Quiz and see if you can master the game's strategy. Learn how to play Fibonacci Nim, explore the Fibonacci Nim rules, and test yourself with engaging Fibonacci Nim trivia. From calculating allowed moves to predicting opponents' steps, you'll pick up strategies that can turn any match in your favor. Whether you're aiming to excel at the Fibonacci Nim game or just sharpen your strategic thinking, this free quiz reveals winning patterns that separate novices from veterans. Curious how you stack up against other puzzle buffs? Boost your brain with our Fibonacci Day Quiz or switch to some bridge trivia to keep your edge sharp. Ready to begin? Dive in and master your move!

In Fibonacci Nim, what limits the maximum number of tokens a player may remove on a turn (after the first move)?
The next Fibonacci number after the previous move
Twice the number removed in the previous move
The same number as the previous move
Half the number removed in the previous move
After the first move, the rule of Fibonacci Nim states that you may remove at most twice the number of tokens your opponent removed on the previous turn. This dynamic limit is what gives the game its name and strategic depth. It forces players to think ahead about how their removal affects future options. Wikipedia
Which sequence do the losing positions (P-positions) in Fibonacci Nim coincide with?
Fibonacci sequence
Prime number sequence
Triangular number sequence
Lucas number sequence
The P-positions in Fibonacci Nim are exactly the Fibonacci numbers. If a heap has a size equal to a Fibonacci number, the player to move is at a theoretical disadvantage. This insight is fundamental to playing optimally. Wikipedia
What theorem guarantees that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers?
Pascal's rule
Cassini's identity
Zeckendorf's theorem
Binet's formula
Zeckendorf's theorem states that each positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers. This representation underpins the winning strategy in Fibonacci Nim. The uniqueness ensures there is a single optimal removal at each step. Wikipedia
On the first move of Fibonacci Nim, what is the restriction on the number of tokens a player can remove?
At least one and at most all but one of the tokens
At least one and at most half of the tokens
Exactly one token
Any positive number up to the entire heap
In Fibonacci Nim the first player must remove at least one token but cannot take the entire heap on the initial move. This ensures the game begins under controlled constraints that tie into the later doubling rule. Subsequent moves then follow the "twice the previous" limit. Wikipedia
If the previous move removed 3 tokens, what is the maximum you can remove on your next turn?
8
6
5
3
Since players may remove up to twice the number of tokens taken in the previous move, if your opponent removed 3 tokens you can remove at most 6 on your turn. This rule guides the flow of the game and keeps it linked to past moves. Wikipedia
Which heap size is a losing (P-position) in Fibonacci Nim?
9
7
10
8
The P-positions in Fibonacci Nim align with Fibonacci numbers, so 8 is a losing position if it's your turn to move. From a heap of 8, any legal removal transfers to a winning (N) position for the opponent. Wikipedia
According to the optimal strategy, after expressing the heap in Zeckendorf representation, you should remove how many tokens?
The sum of the two smallest Fibonacci numbers
The largest Fibonacci number in its representation
The difference between the two largest Fibonacci numbers
The smallest Fibonacci number in its Zeckendorf representation
The optimal strategy is to write the heap size as a sum of non-consecutive Fibonacci numbers and remove the smallest term in that sum. This move transitions the game to a P-position, forcing your opponent into a losing path. Wikipedia
For a heap of size 20, with Zeckendorf representation 13 + 5 + 2, how many tokens should you remove for an optimal move?
3
5
2
13
Given the Zeckendorf representation 13 + 5 + 2, the smallest Fibonacci term is 2. Removing 2 tokens moves the heap to 18, which is a P-position - part of the winning strategy. Wikipedia
For a heap of size 17, which is the optimal number of tokens to remove in Fibonacci Nim?
8
5
1
3
The Zeckendorf representation of 17 is 13 + 3 + 1. Removing the smallest term (1) lands the heap on 16, which is a P-position. This is the core of the optimal play. Wikipedia
Fibonacci Nim falls under which class of combinatorial games?
Cooperative
Partisan
Biased
Impartial
Fibonacci Nim is an impartial game because both players have exactly the same options available from any given position. There is no role-based move restriction, distinguishing it from partisan games. Wikipedia
What key property of the Zeckendorf representation ensures its uniqueness?
The Lucas sequence is used instead
Only even-indexed Fibonacci numbers are allowed
Exactly three Fibonacci numbers must be used
No two consecutive Fibonacci numbers are used in the sum
Zeckendorf's theorem requires that the representation use non-consecutive Fibonacci numbers. This non-consecutiveness is what guarantees that the decomposition is unique for each positive integer. Wikipedia
As n grows large, the ratio of consecutive P-positions in Fibonacci Nim approaches which value?
Pi (3.1416…)
The golden ratio (1.618…)
Euler's number (2.718…)
The square root of 2 (1.414…)
The P-positions of Fibonacci Nim are Fibonacci numbers, and the ratio of consecutive Fibonacci numbers converges to the golden ratio (? ? 1.618). This limit underlies many of the game's mathematical properties. Wikipedia
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Study Outcomes

  1. Understand Fibonacci Nim Rules -

    Learn the core mechanics and rules governing the Fibonacci Nim game, including how moves are determined by Fibonacci numbers.

  2. Apply Fibonacci Constraints -

    Use the Fibonacci sequence to inform your move choices and ensure each turn adheres to game guidelines.

  3. Analyze Game Positions -

    Determine winning and losing positions by evaluating pile sizes and potential moves based on nim theory.

  4. Identify Winning Strategies -

    Recognize patterns and tactics that lead to victory in Fibonacci Nim, from basic maneuvers to advanced strategic plays.

  5. Evaluate and Improve Performance -

    Assess your quiz results to pinpoint strengths and weaknesses, then refine your approach for future games.

Cheat Sheet

  1. Fundamental Rules of Fibonacci Nim -

    In the fibonacci nim game, players alternately remove counters from a single pile, with the first move taking at least one but not all counters and each subsequent move limited to at most twice the previous removal. This rule transforms a static subtraction game into a dynamic challenge that rewards planning. Mastering these fibonacci nim rules gives you a solid foundation for advanced tactics.

  2. Zeckendorf's Theorem and P-Positions -

    Zeckendorf's theorem guarantees every positive integer has a unique representation as a sum of nonconsecutive Fibonacci numbers (e.g., 17 = 13 + 3 + 1), and those Fibonacci numbers themselves (1, 2, 3, 5, 8, 13,…) are the P-positions in fibonacci nim. When the pile size is a Fibonacci number, optimal play by your opponent ensures a win, so recognize these "safe" sizes quickly. A handy mnemonic is "no two in a row" to recall nonconsecutive sums.

  3. Greedy Removal via Fibonacci Representation -

    From any non-P-position in the fibonacci nim game, the winning move is to subtract the smallest Fibonacci number in the Zeckendorf representation of the current total, steering your opponent back to a P-position. For example, from 17 counters (13+3+1), remove 1 to leave 16 and force a losing layout. Remember: "peel off the littlest leaf" to secure the next advantage.

  4. Sprague - Grundy Application -

    As an impartial game, Fibonacci Nim assigns each heap a Sprague - Grundy value g(n), computed via g(n)=mex{g(n−s): 1≤s≤2⋅lastMove} with g(0)=0. Research by Cameron and Fraenkel links these nimbers neatly to Zeckendorf indices, simplifying analysis. Once you tabulate g(n) up to your target pile size, patterns emerge quickly, boosting your fibonacci nim trivia prowess.

  5. Extending to Multiple Heaps -

    In disjunctive sums of fibonacci nim heaps, calculate the binary XOR of all heap nimbers - any nonzero result signals a winning position. This mirrors classic nim strategy: always move to restore an overall XOR of zero. For instance, heaps sized 7 and 10 have nimbers 4⊕5=1, so you're in a winning spot if you play optimally.

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