Half a Lottery, Plus Change

Step 1 — Count the tickets on each tier

Fix the winning sequence. A ticket agrees with it in exactly m positions when m of the seven slots carry the winning number and the remaining 7 − m slots each carry one of the other 18 numbers. Choosing which slots agree and then filling the rest gives:

|S_m| = C(7, m) × 18^{7 − m}

These “shells” partition the ticket space — every one of the 19⁷ tickets sits on exactly one of them — and the three paying shells are:

Readers of coding theory will recognise the shells: they are the sets of sequences at Hamming distance 1, 2 and 3 from the winning one. A lottery prize table is a price list for Hamming distance.

Step 2 — Price the whole table

Here is the one idea the whole puzzle turns on. If you bought every ticket, you would win every prize exactly once. One jackpot, 126 Match-6s, 6,804 Match-5s, 204,120 Match-4s — the entire table, paid one time, for exactly 19⁷ dollars. Call the secondary part of the table T_sec:

T_sec = 126 × $500,000 + 6,804 × $7,500 + 204,120 × $100 = $134,442,000

Spread over the 893,871,739 tickets, that is the secondary return on a single dollar:

r = T_sec / N = $134,442,000 / 893,871,739 = $0.1504

Answer 1: about 15.0 cents of every dollar comes back through the small print. That is not a made-up figure. Powerball’s rules put 50% of sales into prizes and 34% of sales into the jackpot, leaving its eight fixed prizes a combined 16 cents per dollar; the group that bought nearly every Lotto Texas combination in April 2023 collected about $2.5 million in secondary prizes on roughly $25.8 million of tickets — 9.5 cents. Our table is a shade more generous than Texas and a shade less generous than Powerball.

Notice the per-tier form of the same arithmetic: a tier’s return per dollar is |S_m| × W_m / N, which is simply the fraction of its fair prize it pays, where the fair prize is N / |S_m| — the amount at which that tier alone would break even. A Match-6 is “fairly” worth $7,094,220; the $500,000 on offer is 7.0% of that and returns 7.0 cents on the dollar. Match 5 pays 5.7% of fair, Match 4 pays 2.3% of fair. Add them up and you have the 15.0.

Step 3 — What a ticket is worth now

A ticket’s expected winnings are its share of the jackpot plus its share of the secondary table. Buying k distinct tickets at $1 each:

E[profit] = k × (J_net + T_sec) / N − k

With J_net = $378,000,000, the bracket is 0.4229 + 0.1504 = 0.5733, so each ticket is worth 57.3 cents and loses 42.7 cents. Last time it lost 57.7 cents. At the coin-flip purchase of 446,935,870 tickets, the expected secondary take is $67.2 million — roughly 63 Match-6s, 3,402 Match-5s and 102,060 Match-4s, half of every shell — and the expected result improves from a loss of $257.9 million to a loss of $190.7 million. Better. Not good.

Step 4 — The break-even jackpot

Set the bracket to 1. The lottery pays, in expectation, exactly when the whole table is worth the whole ticket space:

J_net + T_sec ≥ N  ⇒  J_net ≥ $893,871,739 − $134,442,000 = $759,429,739

Converting back through the 60% lump sum and the 37% tax, J_adv ≥ $759,429,739 / 0.378:

J_adv ≥ $2,009 million

Answer 2: an advertised jackpot of about $2.01 billion, down from $2.36 billion without the small print. The secondary table is worth $356 million of advertised jackpot — a real number, and still far short of the $1 billion on the billboard.

Step 5 — How generous would the small print have to be?

Hold the jackpot at $1 billion and ask the table to make up the difference:

T_sec ≥ N − J_net = $893,871,739 − $378,000,000 = $515,871,739

The table is worth $134,442,000, so every prize must be scaled by 515,871,739 / 134,442,000:

factor = 3.84

Answer 3: multiply every secondary prize by about 3.84 — Match 6 to $1.92 million, Match 5 to $28,800, Match 4 to $384. The small print would then return 57.7 cents on every dollar, against Powerball’s 16. If the commission preferred to enrich one tier, the Match-6 prize alone would have to rise to $3.53 million: half of its fair value of $7.09 million, where Powerball’s nearest cousin, the $1 million for five white balls, pays 8.6% of fair. No lottery on Earth writes small print like that, because the small print is funded from the same half of sales as everything else.

Bonus — The neighbor

Look again at the expected-profit formula. The k is outside the bracket. Each ticket you hold has the same expected secondary winnings — 15.04 cents — whether you hold one ticket or four hundred million, and each costs the same dollar. The neighbor is right that half the tickets win half the Match-4s; he has forgotten that half the tickets also cost half the money, and that the ratio of the two never moves. The k cancelled in the original puzzle and it cancels again. No purchase size changes any of the three answers. Volume multiplies the small print; it also multiplies the price of admission by exactly the same factor.

The uncomfortable punchline

Fixed secondary prizes are a rebate, not a strategy. They shift the break-even line — here by $356 million of advertised jackpot — but they do not bend it. Expected value per ticket remains a flat line in the number of tickets, and the lottery is profitable if and only if the whole prize table, jackpot plus every secondary prize counted once, is worth more than the price of every ticket. The small print changed the mood. It did not change the economics.

When the small print does bend the line

Fixed prizes are only one way to write secondary rules, and the others behave differently for a large buyer. Use the interactive supplement to watch the curve change shape.

Pools funded from the same draw. Many games pay their middle tiers pari-mutuel: a fixed percentage of that draw’s sales is split among the tier’s winners. Lotto Texas does this for five-of-six, which is why that prize moves from draw to draw — and why, in the 2023 buyout, the group’s 289 five-of-six tickets each paid about $2,015, a perfectly ordinary amount, even though sales were more than ten times normal. The pool grew with the buyer. A pool funded from your own purchases pays you, in expectation, exactly your share of sales. Verdict: a shift, same as fixed prizes.

Pools capped in dollars. Powerball’s rules reserve the right to make its fixed prizes pari-mutuel when a draw’s winners exceed the money set aside for them. A capped pool is the whale’s enemy: buy half the tickets and you hold half the winners of a pool that did not grow to meet you, so your per-ticket return falls as k rises. Verdict: the line bends downward.

Roll-downs. The one documented case of secondary prizes making a lottery profitable is Massachusetts’ Cash WinFall (2004–2012). When its jackpot reached $2 million without a winner, the jackpot “rolled down” into the match-5, match-4 and match-3 tiers. That is a pool funded by previous draws’ sales, dropped into the current one, and it made a $2 ticket worth more than $2 for anyone buying in bulk. Betting groups, including one formed by MIT students, spent roughly $40 million over seven years and took in about $48 million; in August 2010 one group bought about 700,000 tickets for a single draw and cashed 860 of its 963 winning slips. Note what the roll-down rewards. In our lottery, with 100 million public tickets in play and the $378 million rolling down to the lower tiers if nobody hits it, a purchase of 10 million tickets would be worth about $3.61 per dollar — while the half-lottery purchase would still lose 12 cents a ticket, because the whale is mostly sharing the pool with himself. Verdict: the line bends upward, and toward the small buyer.

So the answer to “which secondary rules change the economics?” is: only the ones whose pool is not funded by the tickets you are buying. Prizes paid from this draw’s sales, fixed or pari-mutuel, cannot return more than the fraction of sales the commission set aside. Money from somewhere else can.

Where this leads

Because 19 is prime, the ticket space is the vector space F₁₉⁷, the prize tiers are Hamming shells around the winning point, and a question like “how many tickets guarantee at least a Match-6?” is asking for a covering code of radius 1 over a 19-letter alphabet. Each ticket covers the 1 + 126 = 127 sequences within distance 1 of it, so the sphere-covering bound says you need at least 893,871,739 / 127, which rounds up to 7,038,361 tickets. No set of tickets can tile the space with those 127-sequence balls exactly (19⁷ is not a multiple of 127, and perfect codes over F₁₉ exist only at lengths (19^r − 1) / 18, which 7 is not), so some overlap is forced and the true minimum — unknown, like most values of K_q(n, R) — is at least that and probably more. The “wheeling systems” sold to lottery players are covering designs of exactly this kind. Ellenberg tells the story of how the Cash WinFall syndicates chose their tickets using combinatorial designs related to the Fano plane: a design cannot buy you expected value — that is fixed by the table — but it can make your haul of small prizes nearly deterministic. It buys a quieter night, not a richer one. Guaranteeing a Match-6 costs at least seven million dollars for a prize of half a million, which is the whole puzzle in one sentence.