Game Mechanics and Graph Theory
The five labelled vertices have ten possible edges, the edge set of the complete graph K5. Ten independent fair bits choose which edges are present in the drawn graph. The draw can contain none, some or all of those edges. Its chromatic number is the minimum number of colours needed so that adjacent vertices have different colours. The raw reward is that number minus one.
Modes and Expected Return
There are three distinct modes, distinguished by the power applied to the raw reward. Each mode normalizes the result by the exact mean over all equally likely complete draws, resulting in a 96% expected gross return for all three. Mode 1 raises the reward to the power of 1. Mode 2 raises it to the power of 2. Mode 3 raises it to the power of 3. These modes are mathematical simulations of the same underlying graph structure, not different card or slot reels.
Stakes and Controls
Choose a whole stake of 10, 20, 50 or 100 virtual coins and a reward power of 1, 2 or 3, then draw one board. The stake is debited once at the start. Stake, mode and refill controls stay locked until settlement; sound remains available. There is no automatic queue, stop button or early cash-out.
Settlement and Payouts
The gross payout includes any stake returned; it is not net profit. The demo adds the fractional remainder from the previous payout to the current gross amount, credits the integer part and saves the new remainder separately. A zero-reward graph returns zero coins. The remainder survives settlement, an eligible manual refill and reload.
Wallet and Refill Mechanics
The demo starts with an initial balance of 1,000 virtual coins. If the idle balance drops below 10, you can manually restore it to 1,000. This restoration is not automatic and does not add the reset amount to an existing balance; it sets the balance directly. The fractional carry persists through manual refills and reloads. If you reload the page during an unfinished draw, the round ends without a refund or resume of progress. A saved wallet does not mean a round or all progress resumes automatically.
Mathematical Evidence and Simulation
Separate saved samples of 1,000,000 rounds per mode give estimates close to the calculated expected return. Mode 1 shows an expected RTP of 96% with a measured RTP of 96.06891%. Mode 2 shows an expected RTP of 96% with a measured RTP of 96.00745%. Mode 3 shows an expected RTP of 96% with a measured RTP of 96.01726%. The final carry values vary slightly between modes, reflecting the persistence of fractional changes. These figures represent sample observations over a large number of rounds, not guarantees for short sessions.
Frequently asked questions
Is Chromatic Stair a slot or a card game?
It is an original graph-based virtual-coin demo. Ten bits choose a subgraph of K5; the raw reward is its chromatic number minus one. It uses no reels or cards.
How does the manual refill work?
When your idle balance falls below 10 virtual coins, you can click the refill button. This sets your balance to 1,000 coins. It is a manual action, not automatic. The fractional carry from previous rounds persists into the new balance.
What happens if I reload the page mid-draw?
An unfinished round ends without a refund or resume. You do not get your stake back for the incomplete draw. You must start a new draw with a new stake capture.
