When the card you put underneath comes back
Your data
What makes this countable rather than guessable is that the cards you put underneath go under in the order you put them there. Nothing here is tied to one game: any deck you draw off the top of and add to the bottom of is a queue, not a shuffle.
Results
| Turns for the whole deck to come around | — |
| Turn a card put underneath now comes back | — |
| Turns whose cards all come back on that same turn | — |
| Cards the deck loses each turn | — |
| Turn the deck can no longer pay a full draw | — |
Turn by turn, and when each one comes back
| Turn | Cards in the deck | Turns it waits | Comes back on turn |
|---|
Read the last column downwards and the useful thing appears: it repeats. Several turns in a row send their cards back on the very same turn, because the queue shortens while they queue, and a card put under later catches up with one put under earlier.
So holding a card back for one more turn often changes nothing at all about when you next see it. The choice of what to put underneath is a real one; the choice of when, inside that window, mostly is not.
None of this needs a probability, which is what makes it worth counting. Once the order underneath is known, the question is not how likely a card is to turn up but which turn it turns up on, and that is a thing you can plan against rather than hope for.
Why is this countable at all?
Because what goes underneath goes underneath in order. That makes the deck a queue rather than a shuffle, and a queue has positions you can count instead of odds you have to guess at.
A card you put under now sits below everything already there, so it comes up once those have been drawn: the deck size divided by what you draw each turn, rounded up.
Does it matter which turn I put it under?
Far less than it feels, once the deck is shrinking. The queue in front of the card gets shorter while the card waits, so a card put under later catches up with one put under earlier and they arrive together.
With forty cards, drawing four and putting one under each turn, everything put under on turns two, three, four and five comes back on turn twelve.
| Turn you put it under | Cards in the deck | Turns it waits | Comes back on turn |
|---|---|---|---|
| 1 | 40 | 10 | 11 |
| 2 | 37 | 10 | 12 |
| 3 | 34 | 9 | 12 |
| 4 | 31 | 8 | 12 |
| 5 | 28 | 7 | 12 |
When does that stop being true?
When the deck stops shrinking. If you put under exactly as many as you draw, the queue never gets shorter and each card waits the same number of turns as the one before it.
Then the arrivals spread out one per turn instead of bunching, and choosing the turn really does choose the arrival.
What does the page not know?
Anything that reorders the deck: a shuffle, a search, a card that moves others from the bottom, or drawing an extra card outside the normal turn.
It also treats every turn as identical, so a turn where you draw more or put more under has to be read as a different line rather than fitted into this one.
Reactions
0
0 Comments
Be the first to comment