LotteryWheels.com

The 163-ticket wheel

For a lottery that draws 6 numbers from 49, 163 tickets are the fewest anyone has found that guarantee three matching numbers, whatever six are drawn. The record has stood since November 2000. Nobody knows whether fewer would do: the best proof so far says at least 87 are needed.

Where it came from

For years, wheel enthusiasts competed in the Wheeling Challenge, run online by Richard K. Lloyd from the mid-1990s, to find the smallest 6-from-49 wheel that guarantees three matching numbers. The record fell from 174 tickets to 168 (Uenal Mutlu), then 167, 165 (Salvatore Minacapelli, April 2000) and 164 (Dragan Stojiljković and Rade Belić, July 2000). Stojiljković and Belić reached 163 on 17 November 2000.

Several other people later found 163-ticket wheels of their own. The challenge broke ties by how many three-number combinations a wheel covers, and the version most often quoted today is Peter Rosendahl's from June 2009, which won that tie-break. Nobody has published a 162.

How it's built

Every 163-ticket wheel is two smaller wheels side by side:

  • 77 tickets on 22 of the numbers, forming a Steiner system: every three of those 22 numbers are together on exactly one ticket. It's a famous structure in mathematics, related to the Golay code used in error correction, and 77 is the fewest tickets that can hold every three of 22 numbers.
  • 86 tickets on the other 27 numbers, which match three whenever at least four of the drawn numbers are among those 27.

Any six drawn numbers either include at least three of the 22, and the Steiner ticket holding those three matches them, or at most two of the 22, which leaves at least four among the 27 for the second wheel. The 22-number half can't be improved, so the whole record rests on the 27-number half: an 85-ticket version would make 162. The 86 was found by computer search, and nobody has published an 85.

What it really pays

In Canada's Lotto 6/49, at $3 a play, 163 tickets cost $489 a draw. The guarantee is one three-number match: $10. More is possible, but not guaranteed.

Random tickets come surprisingly close: 163 random Quick Picks include a three-number match 95% of the time, with about 2.9 three-number matches a draw on average, exactly as many as the wheel. The wheel turns 95% into 100%; it doesn't change what you win on average. That's true of every wheel: here's why.

Our own version: 163 tickets, equal to the record

Every wheel on LotteryWheels.com is built by our own software, and so is this one. It uses the same two-part recipe: our own Steiner system on 1–22 (77 tickets) and our own 27-number wheel on 23–49 (86 tickets), which our search found starting from our own wheels, never from anyone else's. That's 163 in all, the same as the record. Our software has checked it against all 13,983,816 possible draws. A search for a smaller 27-number half is still running on our servers; if it finds one, it'll appear here.

Show all 163 tickets
  1. 01 02 03 05 14 17
  2. 01 02 04 13 16 22
  3. 01 02 06 07 19 21
  4. 01 02 08 11 12 18
  5. 01 02 09 10 15 20
  6. 01 03 04 11 19 20
  7. 01 03 06 08 10 13
  8. 01 03 07 09 16 18
  9. 01 03 12 15 21 22
  10. 01 04 05 07 08 15
  11. 01 04 06 09 12 17
  12. 01 04 10 14 18 21
  13. 01 05 06 18 20 22
  14. 01 05 09 11 13 21
  15. 01 05 10 12 16 19
  16. 01 06 11 14 15 16
  17. 01 07 10 11 17 22
  18. 01 07 12 13 14 20
  19. 01 08 09 14 19 22
  20. 01 08 16 17 20 21
  21. 01 13 15 17 18 19
  22. 02 03 04 08 09 21
  23. 02 03 06 12 16 20
  24. 02 03 07 11 13 15
  25. 02 03 10 18 19 22
  26. 02 04 05 06 10 11
  27. 02 04 07 17 18 20
  28. 02 04 12 14 15 19
  29. 02 05 07 09 12 22
  30. 02 05 08 13 19 20
  31. 02 05 15 16 18 21
  32. 02 06 08 15 17 22
  33. 02 06 09 13 14 18
  34. 02 07 08 10 14 16
  35. 02 09 11 16 17 19
  36. 02 10 12 13 17 21
  37. 02 11 14 20 21 22
  38. 03 04 05 12 13 18
  39. 03 04 06 07 14 22
  40. 03 04 10 15 16 17
  41. 03 05 06 09 15 19
  42. 03 05 07 10 20 21
  43. 03 05 08 11 16 22
  44. 03 06 11 17 18 21
  45. 03 07 08 12 17 19
  46. 03 08 14 15 18 20
  47. 03 09 10 11 12 14
  48. 03 09 13 17 20 22
  49. 03 13 14 16 19 21
  50. 04 05 09 14 16 20
  51. 04 05 17 19 21 22
  52. 04 06 08 16 18 19
  53. 04 06 13 15 20 21
  54. 04 07 09 10 13 19
  55. 04 07 11 12 16 21
  56. 04 08 10 12 20 22
  57. 04 08 11 13 14 17
  58. 04 09 11 15 18 22
  59. 05 06 07 13 16 17
  60. 05 06 08 12 14 21
  61. 05 07 11 14 18 19
  62. 05 08 09 10 17 18
  63. 05 10 13 14 15 22
  64. 05 11 12 15 17 20
  65. 06 07 08 09 11 20
  66. 06 07 10 12 15 18
  67. 06 09 10 16 21 22
  68. 06 10 14 17 19 20
  69. 06 11 12 13 19 22
  70. 07 08 13 18 21 22
  71. 07 09 14 15 17 21
  72. 07 15 16 19 20 22
  73. 08 09 12 13 15 16
  74. 08 10 11 15 19 21
  75. 09 12 18 19 20 21
  76. 10 11 13 16 18 20
  77. 12 14 16 17 18 22
  78. 23 24 26 32 46 47
  79. 23 24 29 35 40 41
  80. 23 24 30 36 40 42
  81. 23 24 31 37 40 43
  82. 23 24 38 39 40 46
  83. 23 25 27 28 34 39
  84. 23 25 29 36 43 45
  85. 23 25 30 35 43 44
  86. 23 25 30 37 41 45
  87. 23 25 31 35 42 45
  88. 23 25 33 39 48 49
  89. 23 26 34 38 44 48
  90. 23 27 32 38 44 49
  91. 23 27 33 34 40 45
  92. 23 28 32 33 34 48
  93. 23 28 33 38 44 47
  94. 23 28 40 45 48 49
  95. 23 29 30 31 39 46
  96. 23 35 36 37 39 46
  97. 23 36 38 41 44 45
  98. 23 39 41 42 43 46
  99. 24 25 27 33 38 48
  100. 24 25 28 34 38 49
  101. 24 25 29 30 31 44
  102. 24 25 35 36 37 44
  103. 24 25 41 42 43 44
  104. 24 26 34 39 45 48
  105. 24 27 28 34 40 44
  106. 24 27 32 39 45 49
  107. 24 28 33 39 45 47
  108. 24 29 30 39 43 45
  109. 24 29 36 38 43 46
  110. 24 30 37 38 41 46
  111. 24 31 35 38 42 46
  112. 24 31 36 37 39 42
  113. 24 33 40 44 48 49
  114. 25 26 32 44 45 47
  115. 25 26 34 40 46 48
  116. 25 27 32 40 46 49
  117. 25 28 33 40 46 47
  118. 25 29 35 38 39 41
  119. 25 29 37 38 42 44
  120. 25 30 36 38 39 42
  121. 25 31 37 38 39 43
  122. 25 31 40 41 45 46
  123. 26 27 29 33 37 42
  124. 26 27 30 33 35 43
  125. 26 27 31 33 36 41
  126. 26 27 32 34 47 49
  127. 26 28 29 30 36 49
  128. 26 28 31 42 43 49
  129. 26 28 35 37 41 49
  130. 26 28 36 37 43 47
  131. 26 29 31 35 47 48
  132. 26 30 41 42 48 49
  133. 26 32 38 39 40 47
  134. 27 28 29 31 35 49
  135. 27 28 30 41 42 47
  136. 27 28 34 38 45 46
  137. 27 28 39 44 46 48
  138. 27 29 30 36 47 48
  139. 27 31 42 43 47 48
  140. 27 35 37 41 47 48
  141. 27 36 37 43 48 49
  142. 28 29 32 37 42 48
  143. 28 30 32 35 43 48
  144. 28 31 32 36 41 48
  145. 29 30 31 38 40 45
  146. 29 31 35 36 41 43
  147. 29 32 33 34 41 43
  148. 29 34 37 42 47 49
  149. 29 35 41 44 45 46
  150. 29 36 39 40 43 44
  151. 30 31 32 33 34 37
  152. 30 34 35 43 47 49
  153. 30 35 37 40 42 46
  154. 30 36 42 44 45 46
  155. 30 37 39 40 41 44
  156. 31 34 36 41 47 49
  157. 31 35 39 40 42 44
  158. 31 37 43 44 45 46
  159. 32 33 34 35 36 42
  160. 33 34 39 44 46 49
  161. 33 38 45 46 48 49
  162. 35 36 37 38 40 45
  163. 38 40 41 42 43 45

Open in the playslip guide

Is a whole-game wheel worth playing?

For most players, no. Covering all 49 numbers costs a lot for a small guaranteed prize. A wheel on the numbers you like costs far less, and the site shows exactly how often its guarantee comes into play: make a Lotto 6/49 wheel. If two matching numbers are enough, the smallest whole-game wheel is far cheaper: 19 tickets guarantee two.

Sources: the Wheeling Challenge pages and their record wheels (lottery.merseyworld.com, read from Internet Archive copies); P. C. Li and G. H. J. van Rees, "Lotto Designs", in the CRC Handbook of Combinatorial Designs (2nd ed., 2006): "87 ≤ L(49, 6, 6, 3) ≤ 163 … The upper bound is obtained by adjoining the 77 blocks of the Steiner System, S(3,6,22) to the 86 blocks of a (27,6,4,3)-lotto design". Prices and prizes are from the Lotto 6/49 rules.