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19 tickets that guarantee a Lotto 6/49 prize

Whatever six numbers are drawn, at least one of these 19 tickets matches two of them, and in Lotto 6/49 two numbers win a free play. Nineteen is the fewest that can do it. Mathematicians have known the answer since the late 1990s, and a 2023 paper by David Cushing and David Stewart of the University of Manchester settled the last details of the proof.

What the 19 tickets win

At $3 a play, 19 plays cost $57 a draw. Two matching numbers win a free play, or $5 with the Bonus Number, and three win $10. The guarantee is the smallest of those: one free play, worth a $3 play.

Bigger wins can happen too, but nothing more is guaranteed. And 19 random Quick Picks already include a two-number match 96% of the time, with about 2.5 two-number matches a draw on average, exactly as many as these 19 tickets. The wheel turns 96% into 100%; it doesn't change what you win on average. Like any tickets, they pay back less than they cost.

The same 19 tickets work in any game that draws 6 numbers from 49. In Ontario 49 and Western 649, at $1 a play, two numbers also win a free play, so $19 a draw guarantees one. In BC/49, Atlantic 49 and Québec 49, two numbers win nothing without the Bonus Number, so the guarantee is no use there.

The 19 tickets

Mark these on 19 plays. They cover every number from 1 to 49.

  1. 01 02 03 04 05 06
  2. 07 08 09 10 11 12
  3. 13 14 15 16 17 18
  4. 13 14 15 19 20 21
  5. 16 17 18 19 20 21
  6. 22 23 24 25 28 29
  7. 24 25 26 27 30 31
  8. 26 27 28 29 32 33
  9. 28 29 30 31 34 35
  10. 22 23 30 31 32 33
  11. 24 25 32 33 34 35
  12. 22 23 26 27 34 35
  13. 36 37 38 39 42 43
  14. 38 39 40 41 44 45
  15. 40 41 42 43 46 47
  16. 42 43 44 45 48 49
  17. 36 37 44 45 46 47
  18. 38 39 46 47 48 49
  19. 36 37 40 41 48 49

Open in the playslip guide

Why they always work

Split the 49 numbers into five groups: 1–6, 7–12, 13–21, 22–35 and 36–49. Six numbers can't land in five groups without two of them sharing a group, so it's enough for every two numbers in the same group to be on a ticket together:

  • 1–6 and 7–12: one ticket each, the whole group.
  • 13–21: three tickets. Split the nine into three threes; each ticket holds two of the threes.
  • 22–35 and 36–49: seven tickets each, from the Fano plane, a pattern of seven points and seven lines in which any two points lie on a line. Each point stands for two numbers, and each line of three points makes a ticket of six.

That's 1 + 1 + 3 + 7 + 7 = 19.

Should you play them?

Only if a guaranteed free play is worth $57 a draw to you. The tickets cover every number, so there's no room for numbers you like. A wheel on your own numbers costs less and shows how often its guarantee comes into play: make a Lotto 6/49 wheel. For the famous wheel that guarantees three numbers, see the 163-ticket wheel.

Sources: P. C. Li, Some Results on Lotto Designs (PhD thesis, University of Manitoba, 1999): "For t = 2, the answer is 19"; David Cushing and David I. Stewart, "Applying constraint programming to minimal lottery designs" (2023), Table 3. Our tickets are our own, and our software has checked them against all 13,983,816 possible draws. Every play's Gold Ball number is a separate draw that these numbers don't affect. Prices and prizes are from the Lotto 6/49 rules; always check the official results.