Guide · Rules and restrictions
Secret Santa exclusions: couples, households, and what each rule costs
Every exclusion you add removes draws from the pile, and most tools never say how many. Here is the arithmetic for the rules people actually use, the one rule that decides whether a couples draw is possible at all, and the case where four people in two couples drops from four possible draws to none.
Sealed Draw counts the surviving draws while you type, and names the rule that broke the draw before you press anything. Every figure here was computed, not estimated.
What an exclusion actually does
A Secret Santa draw is a list of arrows: everyone gives exactly one gift, everyone receives exactly one, and nobody points at themselves. Mathematicians call that a derangement. A rule means deleting arrows before the draw runs, and there are only three kinds worth having.
Self-exclusion is always on and is not really a rule; it is what makes the thing a draw. Households are symmetric: if Ana and Ben live together, neither can give to the other. Blocked pairings are directional, which is how last year's result gets recorded — and ticking both ways deletes twice as many arrows as most people intend, which in a small group is the difference between a draw and a dead end.
The price list
The couples rule is the one everybody wants, so here is what it costs. The first column of numbers counts every draw that exists with no rules beyond self-exclusion; the second, what survives when the whole group is paired off into couples. The last is how many shuffles the draw takes on average, which is simply the number of possible orderings divided by the number that pass.
| People | Draws, no rules | Draws, all couples | Share surviving | Shuffles |
|---|---|---|---|---|
| 4 | 9 | 4 | 44% | 6.0 |
| 6 | 265 | 80 | 30% | 9.0 |
| 8 | 14,833 | 4,752 | 32% | 8.5 |
| 10 | 1,334,961 | 440,192 | 33% | 8.2 |
| 12 | 176,214,841 | 59,245,120 | 34% | 8.1 |
Two things are worth reading off that table. A full set of couples still leaves 59 million draws at twelve people, so nobody runs out of options at party sizes. And the surviving share settles around a third rather than falling further, so the couples rule stops getting more expensive once the group can absorb it. With no rules the shuffle count is 2.7 at every size above three, converging on e.
The half-the-group rule
One sentence decides whether a household draw is possible at all: a household can never be more than half the group.
The reason takes ten seconds. Everyone in a household gives to somebody outside it, and each of those outsiders can receive only one gift. So a household of four needs four different people outside it. In a group of six there are two, and no clever arrangement fixes that: a household of four inside six has exactly zero valid draws, which we verified by counting all of them. This is Hall's marriage theorem wearing a party hat, and it is by a wide margin the most common reason a real roster fails.
At the boundary the draw survives but goes rigid. One household of three inside a group of six leaves 36 draws, down from 265, and the three household members must all give to the three outsiders, who all give back. Sealed Draw prints that message with your own numbers filled in, including how many more participants would clear it.
The cliff: four people, two couples, and last year
Small groups fall off the edge without warning, so here is one worked out. Ana and Ben are a couple; Cal and Dee are a couple. Ana and Ben can only give to Cal or Dee, and Cal and Dee only to Ana or Ben. Pick which of the two gets Ana's gift — two ways — and the other is Ben's. Same on the other side. Two times two is four valid draws, out of the nine that exist with no rules.
Now add the rule everybody reaches for next: no repeat of last year. Say last year ran Ana to Cal, Ben to Dee, Cal to Ben, Dee to Ana. Block those four pairings one way and three of the four draws die. The survivor is Ana to Dee, Ben to Cal, Cal to Ana, Dee to Ben — and a draw with one possible outcome is not a draw.
Tick both ways on the same rule and it goes to zero. There were only eight arrows available to begin with, and blocking last year's four in both directions deletes all eight. Sealed Draw catches that before you press anything, but the lesson generalises: below about six people, the no-repeat rule and the couples rule cannot both be strict.
No swap-backs is a constraint, not extra randomness
There is a fourth switch, off by default: no swap-backs, meaning nobody gives to the person who drew them. People turn it on expecting a better-mixed draw. It does the opposite. It deletes draws, and the ones it deletes are all the same shape.
At four people it removes the three draws where two pairs simply swap, leaving six — and those six are exactly the six that form a single loop, which is what the old shortcut of shuffling everyone into a circle produces. At five people, 24 survive out of 44, again exactly the single-loop draws. Only from six people up do the two separate, at 160 against 120. On a small roster this switch does not add mixing; it turns your draw into the circle, which is the bias we refuse everywhere else, so it ships off.
When you do want it, the cost is about a factor of 1.65 in shuffles — roughly six draws in ten survive the extra check, measured at 0.604 for six people against a theoretical 0.607. It is also the one rule the feasibility check cannot certify in advance, because a swap-back depends on the rest of the draw rather than on any single arrow. It holds whenever the fast method succeeds, and is best effort on the fallback.
The rule of thumb, and where it stops working
To predict what your rules cost without counting anything: a draw takes about ek shuffles, where k is how many people each participant is excluded from. Nobody excluded but themselves gives k of 1 and 2.72 shuffles, which is right. Couples give k of 2 and predict 7.4 against a measured 8.2, close enough to plan with.
Publish that bare and it will mislead you. It assumes exclusions are spread thinly, and collapses once a household is a large share of the group: twenty people in two households of ten still have 13.2 trillion valid draws, but finding one by shuffling takes about 185,000 attempts where the rule of thumb predicts 22,026. Sealed Draw switches methods after 20,000 attempts — it builds one valid pairing directly, then randomises it with swaps accepted only when the result still obeys every rule. We cannot show that gives every valid draw an even chance, so the readout labels it constrained rather than uniform.
One thing we decline to build at all. A single loop option — everyone in one chain — is a Hamiltonian cycle question, genuinely different from the matching question our feasibility check solves, and NP-hard once arbitrary exclusions are involved. We could ship a solver that hangs on some rosters, or a feasibility light that guessed. Both are worse than saying no, so we say no.