Tray of Coins Math Specification

Internal game name: tray_of_coins | Display name: "Tray of Gold"

Overview

Tray of Coins is a coin-toss game. N coins (default 30) are tossed onto a tray divided into three regions. The region holding the most coins wins, and players bet on which region wins.

Region	Label (UI)	Description
RED	Bet on Red	Left region
BLUE	Bet on Blue	Center water region
BLACK	Bet on Black	Right region

Coin Distribution

Each coin lands independently in one of the three regions (a multinomial distribution). Per-character landing weights are configurable and normalized to 1:

Character	w_red	w_blue	w_black
Gentleman	0.36	0.28	0.36
Lady	0.40	0.24	0.36

For coin i, P(lands in region r) = w_r. The count in region r follows the joint multinomial Multinomial(N, {w_red, w_blue, w_black}).

Win Condition

Current rule is majority:

winner = region with strictly the greatest coin count

Outcome Probability

The probability that region r wins is the sum of the multinomial probability mass over every distribution where C_r is strictly greater than every other region:

P(r wins) = sum of Multinomial(c_red, c_blue, c_black)
            over all (c_red, c_blue, c_black) where c_r is the strict maximum

The math module computes this exactly (closed-form summation; N is small) and also provides a Monte-Carlo simulator to cross-check.

Indicative values for default weights (exact values come from the math module):

Outcome	Indicative P
Red wins	~42%
Black wins	~42%
Blue wins	~8%
Draw	~2%

Odds Derivation

fair_odds(r) = 1 / P(r wins)
paid_odds(r) = fair_odds(r) * RTP_target          // RTP_target default 0.95

Bet	Fair odds	Paid odds @ 95% RTP (indicative)
Red	1/0.42 ~ 2.38x	~2.26x
Black	1/0.42 ~ 2.38x	~2.26x
Blue	1/0.08 = 12.5x	~11.88x

RTP(r) = P(r wins) x paid_odds(r) = RTP_target for every bet, by construction.

Draw Handling

A draw occurs when no region has a strict majority: coins fall equally into three regions or into any two regions, so no winner can be decided.

draw_resolution	Behaviour
retoss (default)	Re-run the same round: the caller re-invokes play with the same round identity (secret + server_seed) and a fresh client_seed. No new balance deduction.

Multiplier (Premium) Coins

A configurable set of premium coins (default 2x, 5x, 6x) land on regions like normal coins (weighted) and count toward that region's total. If a premium coin lands on the winning region, its value adds to that region's payout multiplier:

total_multiplier(r) = base_multiplier(r) + sum(value of premium coins landing in r)

The math module solves base_multiplier(r) so that the total RTP (base + premium coins

• bonus) equals RTP_target.

Bonus Mini-Game

After a settled round, a bonus round may trigger:

Parameter	Default
bonus_trigger_probability	0.10
bonus_coin_multiplier	25x
bonus_coins_min / bonus_coins_max	1 / 3

bonus_payout = collected_coins x bonus_coin_multiplier x bet. All parameters configurable.

Configurable Parameters

Parameter	Default	Description
total_coins	30	Coins tossed per round
region_weights (per character)	see table	Landing weights, normalized
regions[].is_bettable	true	Whether players may bet on the region
rtp_target	0.95	Return-to-player target per bet
premium_coins	[{2},{5},{6}]	Values added to winning side
bonus_*	see above	Bonus mini-game parameters