Cycle Cabinet Rules: Count Disjoint Nontrivial Cycles

Scope of this independent guide

Lk31.cc is an independent query-information guide with its own Cycle Cabinet browser simulator. Its name and search queries authenticate no casino operator, official domain, app, licence or commercial affiliation. No outside account, catalogue, financial service or regional availability is established here.

The simulator uses original permutation-cycle rules and free virtual coins. It offers no deposits, withdrawals or cash prizes. Its calculations describe this browser software and cannot test an outside business or commercial game.

One permutation and disjoint nontrivial cycles

One spin draws a uniformly random permutation of six labelled positions zero through five. Read each position as pointing to its permuted label. Follow these pointers until returning to the starting position to identify a disjoint cycle, then continue from an unvisited position.

Raw score counts cycles of length at least two. Fixed points, which point to themselves, are excluded. Count cycles rather than moved positions or adjacent label matches. One whole captured stake covers the six-position permutation; there are no separate row shares. All 720 permutations are equally likely.

Fixed points and cycles of different lengths

Identity [0,1,2,3,4,5] consists only of fixed points and scores zero. The permutation [1,0,3,2,5,4] has three two-position cycles and scores three. The permutation [1,2,3,4,5,0] has one six-position cycle and scores one.

The permutation [1,2,0,4,3,5] has a three-position cycle, a two-position cycle and one excluded fixed point; its score is two. A cycle contributes one regardless of its length. Every position belongs to exactly one cycle; the same cycle cannot be counted again from a different starting position.

Exact 720-permutation means and whole-stake credit

There are 720 equally likely permutations of six distinct labels. Choose Low, Medium or High before spinning. Positive raw scores receive square-root, identity or square weighting respectively; zero remains zero. These are weights rather than final payout multipliers.

For each mode calculate the exact mean weight over all 720 permutations, including the zero-score identity. Gross virtual credit equals the whole captured stake times 0.96 times its observed weight divided by that mean. Every mode has 96% exact expected gross return. One whole-permutation return is paid, without separately paid rows.

The stake is deducted before settlement, so gross credit is not net profit. Modes change positive credits after separate normalisation but preserve positive-score frequency. Neither a nontrivial cycle nor 96% expectation guarantees a session balance or a 96% winning chance.

Captured round and the persisted wallet

A new browser wallet starts with 1,000 free virtual coins. Settled balance and fractional-credit carry persist in localStorage. Reload does not replenish settled funds. Fractions accumulate before whole coins are credited rather than being discarded per spin.

A spin captures one total stake and mode for its permutation. Settings and refill lock during settlement and an active automatic queue. Auto runs at most ten rounds; Stop prevents future rounds while an active spin settles. Optional manual refill is available only while idle with balance below ten. It sets balance to 1,000 and preserves carry. Coins cannot be purchased, transferred, redeemed or withdrawn.