Guide · Rules and restrictions

Secret Santa exclusions: couples, households, and what each rule costs

Every exclusion you add removes possible draws. Here are the counts for couples, households and last year's pairs, the one rule that decides whether a couples draw is possible at all, and the case where four people in two couples go from four possible draws to none.

Ten people, no rules1,334,961 draws Ten people, five couples440,192 draws Largest householdhalf the group

Sealed Draw counts the surviving draws while you type, and names the rule that broke the draw before you press anything. Every figure on this page was computed.

Set up a draw with rules

What an exclusion 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 deletes arrows before the draw runs, and three kinds are worth having.

Self-exclusion is always on; it is what makes the list a draw at all. Households are symmetric: if Ana and Ben live together, neither can give to the other. Blocked pairings are one-way, which is how you record last year's result. Ticking “both ways” deletes twice as many arrows as most people intend, and in a small group that is the difference between a draw and a dead end.

The price list

Here is what the couples rule costs. The first column of counts is every draw that exists with no rules beyond self-exclusion. The second is what survives when the whole group is paired off into couples. The last column is how many shuffles a draw takes on average: the number of possible orderings divided by the number that pass.

Verified counts. Couples means every person is in a two-person household.
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

At twelve people, a full set of couples still leaves 59 million draws, so nobody runs out of options at party sizes. The surviving share settles around a third, 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

A household draw has one hard limit: a household can never be more than half the group.

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. A group of six has only two, so a household of four inside six has exactly zero valid draws; we checked by counting all of them. It is a special case of Hall's marriage theorem, and it is the most common reason a real roster fails. When your roster breaks it, Sealed Draw shows the rule with your own numbers filled in, including how many more participants would clear it.

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.

The cliff: four people, two couples, and last year

Small groups run out of draws quickly, so here is one worked out. Ana and Ben are a couple; Cal and Dee are a couple. Ana and Ben each give to Cal or Dee, and Cal and Dee each give to Ana or Ben. Ana's gift can go to either of the two (two ways), and Ben gives to the other. The same holds on the other side. Two times two is four valid draws, out of the nine that exist with no rules.

Now add a no-repeat rule for 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 are gone. The one left is Ana to Dee, Ben to Cal, Cal to Ana, Dee to Ben, and anyone who knows the rules can predict it.

Tick “both ways” on the same rule and the count 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. Below about six people, the no-repeat rule and the couples rule cannot both be strict.

No swap-backs is a constraint that removes draws

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, but all it does is delete draws, and every draw it deletes has two people giving to each other.

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: the result of the old shortcut of shuffling everyone into a circle. 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. So on a small roster the switch turns your draw into the circle, the bias Sealed Draw avoids everywhere else, and that is why it starts unticked.

If you do want it, the cost is about a factor of 1.65 in shuffles: roughly six draws in ten survive the extra check. For six people the exact figure is 0.604; 0.607 is the large-group limit. It is also the one rule the feasibility check cannot certify in advance, because a swap-back is made of two arrows together and the check works on single arrows. The draw you get when you press Seal obeys it. If the fallback method cannot find a draw without swap-backs, the page says so and asks you to turn the rule off or loosen another one.

The rule of thumb, and where it stops working

To estimate what your rules cost without counting anything: a draw takes about e^k 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 correct. Couples give k of 2 and predict 7.4 against a measured 8.2, close enough to plan with.

The estimate assumes exclusions are spread thinly, and it breaks down 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, keeping each swap only when the result still obeys every rule. We cannot show that this gives every valid draw an equal chance, so for those draws the Method readout says “constrained” and “not provably uniform”.

Frequently asked questions

How large does a group need to be for a couples rule to work?

At least twice the size of your largest household. Everyone in a household gives to someone outside it, and each of those outsiders can receive only one gift, so a household of k people needs k people outside it. Four people in two couples is the smallest case that works at all: 4 valid draws out of the 9 that exist with no rules. A household of four inside a group of six leaves zero. Six people in three couples gives 80.

Why does the tool say my rules make a draw impossible?

Because your rules leave no valid draw. Every change to a name or a rule runs a matching check that always finishes and never draws anything, so you get the answer before you press Seal. Five things cause it: a household larger than half the group, someone excluded from everybody, someone nobody may give to, two people with no swap-backs switched on, and the case where several rules overlap. For that last one it names the smallest set of people whose combined rules leave them too few recipients, and the rules involved.

Can I block last year's pairings as well as couples?

Yes, in one direction or in both, and both is where small groups get caught. Four people in two couples starts with 4 valid draws. If last year ran Ana to Cal, Ben to Dee, Cal to Ben and Dee to Ana, blocking those four one way leaves exactly one draw, so the result is fixed and any of the four can work out the rest. Block the same four both ways and nothing is left: zero.

Do exclusions make the draw less random?

No. They make the set of possible draws smaller, and whenever the Method readout says uniform, every draw left in it is equally likely. With five couples in a group of ten, each of the 440,192 permitted draws has the same chance, and the 894,769 your rules removed are simply gone. What exclusions do skew is individual odds: if one person is excluded from four others while everybody else is excluded from one, that person's remaining recipients each become likelier, and no draw method can undo that.

Can everyone be in one big chain, so the gifts form a single loop?

No. Sealed Draw has no single-loop option. Deciding whether a single loop exists under arbitrary exclusions is the Hamiltonian cycle problem, which is harder than the matching question the feasibility check answers and NP-hard in general. Any version of it would either hang on some rosters or show a feasibility light that was guessing. Forcing one loop also biases the draw: it is the party circle shortcut, which at ten people reaches only about 27 percent of the valid draws.