What counts as a rooted tree
Directed Root Trees has four vertices, numbered 0, 1, 2 and 3. Twelve independent fair switches decide whether each directed edge is present: 1 means present and 0 means absent. For example, 0→1 and 1→0 are separate edges. The terminal score counts spanning trees directed outward from root 0. Each of vertices 1, 2 and 3 has exactly one incoming tree edge, and all three can be reached from 0 along the tree.
All twelve edges present give 16 possible rooted trees. If only 0→1, 0→2 and 0→3 are present, there is exactly one: the three-edge star. A cycle disconnected from root 0 does not count. Extra graph edges are allowed; the score counts valid tree subsets, not just whether the whole graph looks like a tree.
Reveal order and conditional quotes
The order is 0→1, 0→2, 0→3, 1→0, 1→2, 1→3, 2→0, 2→1, 2→3, 3→0, 3→1, 3→2. Each unrevealed edge remains equally likely to be present or absent when calculating the current expected tree count.
The gross multiplier is 0.97 × the expected final tree count given the revealed switches, divided by the initial expected count of 2. The opening quote is 0.9700×; manual collection becomes available after the first reveal. The quote can rise or fall. Zero ends the round immediately. If the round reaches reveal 12 with a positive quote, that quote settles automatically.
Manual and Auto collection
Choose By hand or Auto 1.2, 2, 3 or 5 before starting. Auto collects the full current quote on the first reveal that reaches its threshold. It does not cap the return at the threshold. At each reveal, zero is checked first; an earlier completed collection is unaffected by later hidden switches. If the threshold is never reached, the round ends at zero or settles the surviving quote at reveal 12.
With By hand selected, use Collect current quote after a reveal. A click processes any reveals already due before collecting, so it can encounter an automatic ending first. The displayed quote is rounded to four decimal places; settlement uses the full value.
Three worked graph outcomes
A present first edge gives 1.455×, enough to settle Auto 1.2. If all three outgoing edges from root 0 are absent, no rooted tree is possible and the quote reaches zero at reveal 3. If all twelve edges survive as present and the round is still active, the terminal score is 16 and the terminal quote is 7.76×. Earlier Auto collection can finish that same hidden graph before its terminal quote.
Expected return and whole-coin settlement
For collection decisions based only on revealed information, with the round allowed to settle under these rules, expected gross return before whole-coin rounding is exactly 97% of the stake. This is an average across possible rounds, not a promised result for a session. The 0.97 factor is applied once in the quote; there is no second 3% collection deduction. Whole-coin credits include carry between rounds, so an individual wallet credit need not equal 97% of its stake.
A new wallet starts with 1,000 virtual coins. Settlement multiplies the captured stake by the full quote, adds the saved fractional remainder, and credits only the whole-number part. The new remainder is saved for a later settlement; it is never credited separately. For example, stake 20 at 1.455× with carry 0.4 produces 29.5: 29 coins are credited and 0.5 is retained.
Stake, wallet and reload rules
The stake options are 10, 20, 50 and 100 virtual coins. A fresh page starts with stake 20 and By hand selected. Starting a round deducts that stake once and fixes it for the round. Stake and Auto settings, and balance restoration, are locked while the round is active. Sound remains available. Settings stay selected between completed rounds, but reload returns them to the defaults.
When no round is active and the balance is below 10, Restore balance sets it to exactly 1,000 free coins. It does not add 1,000, run automatically or reset the fractional remainder. A balance of exactly 10 is not eligible.
The wallet is saved when the stake is deducted and when returns are credited. Reloading during an unfinished round loses that round and its already deducted stake: there is no refund or resume. The saved wallet and fractional remainder remain if browser storage is retained. Reload resets stake to 20 and Auto to By hand.
Frequently asked questions
Does 97% mean I will get 97 coins back from every 100?
No. It is the exact expected gross return across possible rounds before integer wallet credits and carry. A single round can return zero, less than its stake or more than its stake.
What if Auto never reaches its threshold?
Zero ends the round as soon as it occurs. Otherwise, if no earlier collection occurs, the surviving full quote settles after reveal 12.
Can a later zero cancel an earlier Auto payment?
No. Once Auto collects at a first crossing, that round is settled. Later hidden edges no longer affect its payment.
