Hopscotch / Math sheet

Hop & Cash · 100-step model · Reviewed 27 October 2026

Server-backed virtual-credit demo. Whole-credit stakes, minimum 1. Not a certification or real-money approval.

Model and payout

The multiplier at step k is rounded to two decimals: M(k) = round2(1.10 × 1.16^(k−1)). The first hop succeeds with probability 0.97 / M(1); later hops use M(k−1) / M(k). Thus the ideal cumulative chance to reach step k is 0.97 / M(k). A successful step 100 automatically cashes out.

Before settlement rounding, each fixed cashout step has 97% theoretical RTP. Hit frequency means reaching the chosen step, not making a net profit. Larger target steps have lower hit frequency and greater payout variance.

Rounding and supported stakes

DEMO credits use two decimals. Stakes and payouts are integer minor units. For stake b in minor units and q = 100 × M(k), the payout is floor((b × q + 50) / 100): nonnegative half-up rounding once at settlement. The exact ideal rounded RTP is 97 × payout / (q × b).

All accepted stakes are whole credits, with a minimum of 1. For b = 100 × n minor units, payout = n × q is integral at every step. Settlement rounding therefore introduces no deviation from the ideal 97% RTP. Fractional-credit stakes are rejected before debit; fractional balances and payouts remain supported.

Tests cover denominations with 0, 2 and 3 decimals; only 2-decimal DEMO credits with whole-credit stakes are enabled. Earlier review documents considered fractional stakes, which are not accepted by this release. Finite 53-bit server random draws are distinguished from ideal rational probabilities.

Decision window

The reviewed backend uses three seconds plus a configurable grace period (250 ms by default), shown by the client timer. Expiry automatically cashes the confirmed current multiplier; it does not cause a loss. An accepted jump is never rerolled based on timing. Expiry may select cashout instead of a new jump. For accepted whole-credit stakes, ideal expected return is unchanged by an admissible stopping policy; realized outcomes can differ.

Verification

The dated review includes 120,000,000 simulated full paths across three seeds, five stopping strategies, distributions and z-scores. Simulation supports the analysis; it does not replace exact calculation. Step 100 is rare (about 14.66 expected hits per 40 million paths), so no universal ±0.05 percentage-point simulation guarantee is claimed.

Step distribution

Ideal cumulative hit probability and rounded RTP for a 1-credit preset stake. All five preset stakes have the same RTP shown here.