Two mathematicians from Manchester University have shown how you can always win a prize in Lotto, no matter which balls are drawn.

Their claim and ingenious and elegant solution appeared in the Daily Mail on 1 August 2023.

With more than 45 million combinations of 6 from 59 balls, how is it possible to guarantee to win a Lotto prize?

To pick the numbers that will be drawn randomly is a tall order. There are formidable odds against you winning for sure. Success requires lateral thinking.

My scientific curiosity was piqued when I first noticed the article in the Daily Mail through the MSN news channel. I was determined to examine the contents closely and add my perspective. Here are the results of my investigations into the Manchester method.

It reminded me of a similar scheme that caught my attention many years ago when Lotto played with 49 balls; it sparked my enduring interest in the National Lottery. The Liverpool Hope method, using the Steiner system derived from group theory, guaranteed to win at least two Match 3 prizes for a stake of £290.

Estimated reading time: 33 minutes

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Their Discovery

Dr David Cushing and Dr David Stewart, mathematicians in the Mathematics Department of Manchester University, found that you must play a specific set of 27 boards (not Lucky Dips) to be assured of a prize. Initially, they predicted that 26 boards would suffice, but they didn’t. The reason remains a puzzle. Twenty-seven boards are required for success – see the table below.

Read: How you can GUARANTEE a National Lottery win by buying 27 tickets for more information. (Daily Mail article)

Read: How many lottery tickets do you need to buy to guarantee a win? Manchester’s mathematicians find the answer for more information. (Manchester University article)

Read: Winning the lottery with the Fano plane for more information. (A Piece of the pi: mathematics explained article)

However, there is a catch; you might not make a profit, even though you always win a prize!

All becomes clear when you see Lotto’s prize structure in the table below.

Lotto Prizes

Category	Prize	Remarks
Match 2	Free Lucky Dip[1]	If the rollover limit has been reached or Camelot has declared that Lotto must be won, Match 2 winners receive a cash prize of £5 and a Free Lucky Dip if no one wins the jackpot.
Match 3	£30	The jackpot is shared among the lower prize tiers in a rolldown according to a pre-determined formula. There is a rolldown when the rollover limit has been reached, or Camelot declares that Lotto must be won.
Match 4	£140
Match 5	£1,750
Match 5 plus	£1,000,000
Match 6	Variable	There is a minimum of £2,000,000 on Wednesdays and £3,800,000 on Saturdays. Jackpots depend ultimately on ticket sales. Occasionally, Camelot promotes Lotto by boosting the jackpot to guarantee a larger amount than sales justify. e.g. £20,000,000.

One of the prizes is a Free Lucky Dip. Furthermore, you are more likely to win this prize than any other. It hinders our ambitions since it has no cash value except as a pre-paid entry in a subsequent draw and poleaxes our expectations of triumphal success.

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Their Method

Background

Let’s begin by casting our minds back to our school days. Back then, you might recall Euclidean geometry^[2] with its axioms^[3] and theorems^[4], dealing with properties such as distance, shape, size, relative positions of figures and curves, and entities such as planes, angles and surfaces.

In Euclidean geometry, a point has no length, width or thickness and marks an exact location in space.

However, Euclidean geometry is not the only geometry. In modern mathematics, a point is an element of a set called a space.

The mathematical method employed by the two Manchester mathematicians is based on finite geometry, in which combinations of balls are visualised as points in geometrical shapes.

Finite geometry has a finite number of points. Note that Euclidean geometry is not finite, since a straight line can be considered to consist of an infinite number of points.

The mathematicians used Fano planes to obtain their solution.

Fano plane

A Fano plane is a finite projective plane with the least number of points and lines and the following properties:

• 7 points and 7 lines.
• There are 3 points on every line.
• There are 3 lines through every point.

Notes

• The equilateral triangle contains an inscribed circle whose centre coincides with the centre of the triangle. 
• The sides of the triangle are tangents to the circle.
• The 7 points consist of the three vertices of the triangle, the centre of the circle/triangle, and the three points of tangency that correspond to the midpoint of each side.
• There are seven lines: the three sides of the triangle, three medians^[5], and finally, the circle itself, considered a closed line with its two ends joined up.

Read: A Few of My Favorite Spaces: The Fano Plane for more information about the Fano plane^[6] (an article in the journal Scientific American).

The Manchester Method visualised

You need two triangles and three Fano planes to generate the 27 boards required to guarantee a Lotto win.

To win the lottery, proceed as follows.

• Use pairs of balls in the first triangle: (1, 2), (3, 4), (5, 6) and (7, 8), labelling the centre and the three vertices.
• On the second triangle, label the three vertices (9, 10, 11), (12, 13, 14), and (15, 16, 17), leaving 42 unused balls.
• Beginning with the first Fano plane, assign pairs of balls sequentially, and label each of the 7 points on the plane. i.e. (18, 19), (20, 21), (22, 23), (24, 25), (26, 27), (28, 29) and (30, 31).
• Repeat for the second Fano plane. i.e. (32, 33), (34, 35), (36, 37), (38, 39), (40, 41), (42, 43) and (44, 45).
• Then repeat the process for the third Fano plane. i.e. (46, 47), (48, 49), (50, 51), (52, 53), (54, 55), (56, 57) and (58, 59), exhausting the list of balls and using each ball once.

The next step is to combine the doublets (e.g. 1, 2) and triplets (e.g. 9, 10, 11).

Let’s begin with the first triangle, which has 4 points. Call the points: A (1, 2), B (3, 4), C (5, 6) and D (7, 8).

Mathematics tells us that there are ⁴C₃ (i.e. 4) ways of combining three items from 4 to form our board sextets, namely: 

1. ABC (1, 2, 3, 4, 5, 6)
2. ABD (1, 2, 3, 4, 7, 8)
3. ACD (1, 2, 5, 6, 7, 8)
4. BCD (3, 4, 5, 6, 7, 8)

However, we must try to avoid redundancy (i.e. excessive duplication) to limit the number of boards we need, especially among triplets that generate the lowest cash prize), meaning that we can discount BCD (3, 4, 5, 6, 7, 8) and retain only ABC, ABD and ACD.

Using similar arguments and logical steps, we can process the triangles and three Fano planes to conclude with the set of 27 boards.

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My Analysis

In a Lotto game using 59 balls, Camelot can draw any of the 53 × ⁵⁹C₆ combinations, including the bonus ball. (i.e. 2,388,046,122).

Checking the boards played against all combinations using my bespoke computer software^[7] reveals 72 unique outcomes, summarised in the table below. All win prizes.

Outcomes of the 27 boards, whichever balls are drawn.

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
51	2	0	0	0	0	0	611,250,696	25.596%	2
33	3	0	0	0	0	0	457,275,308	19.149%	3
30	4	0	0	0	0	0	375,437,160	15.722%	4
58	1	0	0	0	0	0	235,599,840	9.866%	1
32	3	1	0	0	0	0	165,307,212	6.922%	4
49	2	1	0	0	0	0	123,534,096	5.173%	3
57	1	1	0	0	0	0	85,407,168	3.576%	2
28	5	0	0	0	0	0	63,127,134	2.643%	5
27	6	0	0	0	0	0	48,824,289	2.045%	6
59	0	1	0	0	0	0	47,618,592	1.994%	1
19	2	2	0	0	0	0	46,348,182	1.941%	4
25	4	1	0	0	0	0	32,134,536	1.346%	5
31	5	1	0	0	0	0	16,328,664	0.684%	6
17	3	2	0	0	0	0	12,903,645	0.540%	5
20	1	2	0	0	0	0	12,572,448	0.526%	3
56	2	0	1	0	0	0	11,412,702	0.478%	3
18	4	2	0	0	0	0	8,109,954	0.340%	6
29	7	0	0	0	0	0	5,011,839	0.210%	7
23	3	1	1	0	0	0	4,840,596	0.203%	5
47	0	2	0	0	0	0	4,238,304	0.177%	2
62	3	0	1	0	0	0	3,186,837	0.133%	4
50	4	0	1	0	0	0	2,827,656	0.118%	5
21	3	3	0	0	0	0	1,700,664	0.071%	6
46	0	4	0	0	0	0	1,603,674	0.067%	4
44	1	1	1	0	0	0	1,480,608	0.062%	3
11	2	2	1	0	0	0	1,207,764	0.051%	5
37	6	1	0	0	0	0	1,068,480	0.045%	7
24	5	0	1	0	0	0	926,334	0.039%	6
48	0	3	0	0	0	0	899,304	0.038%	3
55	0	1	1	0	0	0	881,496	0.037%	2
13	0	2	1	0	0	0	845,244	0.035%	3
35	5	2	0	0	0	0	801,360	0.034%	7
26	8	0	0	0	0	0	624,393	0.026%	8
43	2	1	1	0	0	0	369,198	0.015%	4
22	1	3	0	0	0	0	309,096	0.013%	4
60	4	0	0	1	0	0	294,840	0.012%	5
36	9	0	0	0	0	0	293,090	0.012%	9
8	1	0	2	0	0	0	225,144	0.009%	3
70	2	3	1	0	0	0	213,696	0.009%	6
12	1	2	1	0	0	0	177,444	0.007%	4
16	2	3	0	0	0	0	148,824	0.006%	5
52	6	0	1	0	0	0	80,136	0.003%	7
72	3	4	0	0	0	0	71,232	0.003%	7
42	0	1	2	0	0	0	64,872	0.003%	3
71	5	0	2	0	0	0	53,424	0.002%	7
68	1	2	2	0	0	0	53,424	0.002%	5
66	3	2	0	1	0	0	52,416	0.002%	6
34	7	0	1	0	0	0	50,085	0.002%	8
53	1	1	0	1	0	0	46,800	0.002%	3
7	2	0	2	0	0	0	44,361	0.002%	4
5	0	1	1	1	0	0	34,632	0.001%	3
45	4	1	1	0	0	0	26,712	0.001%	6
65	1	4	1	0	0	0	26,712	0.001%	6
10	0	0	3	0	0	0	23,108	9.68 × 10-4 %	3
41	0	2	0	1	0	0	15,912	6.66 × 10-4 %	3
14	3	2	1	0	0	0	13,356	5.59 × 10-4 %	6
61	4	0	0	0	1	0	5,670	2.37 × 10-4 %	5
64	3	0	3	0	0	0	4,452	1.86 × 10-4 %	6
9	4	0	2	0	0	0	3,339	1.40 × 10-4 %	6
6	3	0	2	0	0	0	1,431	5.99 × 10-5 %	5
69	6	0	0	0	0	1	1,113	4.66 × 10-5 %	7
67	3	2	0	0	1	0	1,008	4.22 × 10-5 %	6
15	1	4	0	0	0	0	954	3.99 × 10-5 %	5
54	1	1	0	0	1	0	900	3.77 × 10-5 %	3
4	0	1	1	0	1	0	666	2.79 × 10-5 %	3
39	0	0	2	1	0	0	624	2.61 × 10-5 %	3
3	0	0	1	2	0	0	612	2.56 × 10-5 %	3
40	0	2	0	0	1	0	306	1.28 × 10-5 %	3
1	0	0	2	0	0	1	159	6.66 × 10-6 %	3
63	0	2	0	0	0	1	159	6.66 × 10-6 %	3
2	0	0	1	1	1	0	24	1.01 × 10-6 %	3
38	0	0	2	0	1	0	12	5.03 × 10-7 %	3

The columns Match 2, 3, 4, etc indicate the number of prizes in those categories. I have arranged them in descending order of likelihood (probability).

Frequency of outcomes

Number of outcomes including Match 2, Match 3, etc

Frequency	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6
0	18	30	39	64	65	69
1	12	17	21	7	7	3
2	9	16	10	1	0	0
3	12	5	2	0	0	0
4	8	4	0	0	0	0
5	5	0	0	0	0	0
6	4	0	0	0	0	0
7	2	0	0	0	0	0
8	1	0	0	0	0	0
9	1	0	0	0	0	0
Total:	72	72	72	72	72	72

The burning question is: Does this discovery benefit us, or does it just raise false hopes?

Let us review the findings.

Reviewing the outcomes

1. I calculated each outcome’s probability from its frequency of occurrence tabulated by my software and expressed the probability as a percentage (0% to 100%) instead of the usual 0 to 1.
2. The minimum number of prizes is 1, and the maximum is 9.
3. Most outcomes have prizes in more than one category.
4. All outcomes win a prize, but not all prizes are paid in cash. Winners matching two balls receive a free Lucky Dip in the following Wednesday or Saturday draw unless the rollover limit has been reached or Camelot has designated the draw as must be won, after which they receive an additional £5 in cash if there are no jackpot winners.
5. Outcomes appearing in red (9) win only Match 2 prizes – Lucky Dips. They account for 1,797,443,749 of the 2,388,046,122 combinations. i.e. 75.27% – more than three-quarters. Ouch! Due to uncertainty, we discount that the Free Lucky Dips could produce cash prizes in a later draw.
6. Sixty-three outcomes win cash prizes. They account for 590,602,373 combinations out of 2,388,046,122. i.e. 24.73%. Multiple prizes are available, and the most likely (at 6.92%) is three Match 2 plus a Match 3, netting us at least £30 – possibly more if there is a rolldown.
7. Despite the firm win guarantee, the results make for grim reading – better wait to order your new luxury car; you might be disappointed!

Likely scenario over one year

What might happen over a year in 104 draws? Let us see.

Firstly, let’s work out the cost.

Buying 27 boards will cost £54 per draw, giving an annual total of £5,616. That is a tidy sum.

Looking at the prize schedule, at the very least, we will need three Match 5 prizes of £1,750, plus several minor ones, to break even, let alone make a profit. Achieving this is a tall order!

We shall:

• consider Free Lucky Dips and cash prizes separately,
• use the Binomial Distribution Function^[8] to calculate the probability of winning between 0 and 104 times,
• and produce graphs with significant probability (>1%).

Here are the results, shown side by side.

There is no point in showing graphs relating to the highest prize categories. i.e. Match 5, Match 5 plus bonus and Match 6. The only realistic possibilities are zero and one! See the table below.

Likely number of successes in the higher prize categories in one year and their probabilities

Likely wins	Match 5	Match 5+	Match 6
0	98.077%	99.963%	99.994%
1	1.905%	0.037%	0.006%>
2 or more	0.018%	>0.000%	>0.000%
Sum:	100%	100%	100%

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Conclusion

Although choosing 27 boards proposed by the Manchester mathematicians guarantees a win every time, there is no guarantee of a profit over a year.

The most likely outcome is to win Free Lucky Dips on 79 occasions and cash prizes on the other twenty-five – cold comfort, indeed!

The chances of a substantial win or even making a profit look slim. If you are lucky, your best hope is that some cash prizes belong to the higher tiers, which is unlikely but still possible. Refer to Miscellaneous in the Appendix for further details.

By all means, play this system, but do so with your eyes open. Major success is unlikely; nursing a loss is more probable.

Finally, if you choose to play the Manchester method, you should avoid following the herd playing with the 27 boards listed at the start of this article.

Instead, you can play with a unique set of 27 boards by remapping each of the fifty-nine balls to different numbers randomly and substituting accordingly. The amended set will have the same outcomes.

Here is a set I generated. However, you can choose your own randomly using the button beneath the table.

An alternative set of 27 Boards guaranteed to win Lotto

Board	Balls
1	1	5	34	46	49	59
2	5	36	46	49	56	59
3	1	34	36	46	56	59
4	17	18	28	32	37	41
5	3	20	23	32	37	41
6	3	17	18	20	23	28
7	22	26	33	35	40	57
8	11	25	26	31	33	55
9	15	26	33	43	47	58
10	11	15	40	43	55	57
11	25	31	40	47	57	58
12	11	22	35	47	55	58
13	15	22	25	31	35	43
14	2	24	51	52	53	54
15	6	14	24	45	50	54
16	24	27	29	42	44	54
17	2	6	27	44	45	51
18	2	14	29	42	50	51
19	6	29	42	45	52	53
20	14	27	44	50	52	53
21	4	9	12	13	30	39
22	7	9	10	16	38	39
23	8	9	19	21	39	48
24	4	7	8	19	30	38
25	4	10	16	21	30	48
26	7	12	13	21	38	48
27	8	10	12	13	16	19

Choose 27 boards yourself

Whatever you do, good luck!

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Notes

• [1] Some people dispute whether this is a real prize! It seems little more than a sop to players to give the illusion of a better chance of winning when the odds of winning a cash prize have worsened since the introduction of 59 balls.
• [2] Euclid was an ancient Greek mathematician regarded as the Father of Geometry, active in Alexandria, Egypt, around 300 BCE. Although little is known about him, his work has influenced mathematics in the Western world and the Middle East from antiquity until the early nineteenth century. Euclidean geometry continues to be taught and studied even to this day.
• [3] Axioms are self-evident truths. e.g. the shortest path between two points is a straight line.
• [4] Theorems are statements deduced from other propositions. e.g. Pythagoras theorem relating to right-angled triangles and its converse, and Apollonius theorem.
• [5] A median of a triangle is a line between a vertex and the midpoint of its opposite side, thus bisecting that side. A triangle has three medians, and they intersect at the same point, the centroid.
• [6] The author, Evelyn Lamb, hints at using the Fano plane to study lotteries, but the type of lottery the author refers to is the Standard Numbers Game.
  Read: Types of Lottery: Standard Numbers Game for more information.
• [7] Fortunately, the costs of this kind of research are not prohibitive. You do not need costly equipment – only time, dedication, ingenuity, peace and quiet, and a personal computer.
  Excel and Word from Microsoft Office have played their parts – in analysis and word processing, respectively.
  I used Borland Delphi for the computer programs. Of those available, no other compiled computer language matched Delphi’s execution speed, which was essential, considering the enormous number of combinations involved in analyses and simulations.
  Borland Delphi uses the Object Pascal language. Embarcadero now own Borland.
• [8] The Binomial Distribution Function, BINOMDIST, built into Microsoft Excel, facilitates the calculations by typing the following formula into a worksheet cell.
  For n successes in 104 trials, each having a probability of p, the formula is:
  =BINOMDIST(n, 104, p, FALSE)
  The last parameter, FALSE, prevents the function from producing a cumulative total.
  The table below shows the probability for each case.
  

  
  

  Probability of success for each prize category in a single draw

  	Free Lucky Dips	Cash Prizes	Match 3	Match 4	Match 5	Match 5+	Match 6	Sum
  Frequency	1,797,443,749	590,602,373	561,106,389	29,040,131	445,836	8,586	1,431	2,388,046,122
  Probability (%)	75.268	24.732	23.496	1.2161	1.8670 × 10-2	3.5954 × 10-4	5.9924 × 10-5	100
  Probability	0.75268	0.24732	0.23496	0.012161	1.8670 × 10-4	3.5954 × 10-6	5.9924 × 10-7	1

  
  

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References

• pure.manchester.ac.uk/ws/portalfiles/portal/272368587/2307.12430v1.pdf
• https://research.manchester.ac.uk/en/publications/you-need-27-tickets-to-guarantee-a-win-on-the-uk-national-lottery

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Appendix

The following tables relate to several prize categories and provide detail for further research and scrutiny.

Free Lucky Dips

Outcomes of the 27 boards, awarding Free Lucky Dips

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
51	2	0	0	0	0	0	611,250,696	25.596%	2
33	3	0	0	0	0	0	457,275,308	19.149%	3
30	4	0	0	0	0	0	375,437,160	15.722%	4
58	1	0	0	0	0	0	235,599,840	9.866%	1
28	5	0	0	0	0	0	63,127,134	2.643%	5
27	6	0	0	0	0	0	48,824,289	2.045%	6
29	7	0	0	0	0	0	5,011,839	0.210%	7
26	8	0	0	0	0	0	624,393	0.026%	8
36	9	0	0	0	0	0	293,090	0.012%	9
Total (out of 2,388,046,122) :							1,797,443,749	75.27%

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Cash Prizes

Outcomes of the 27 boards, awarding cash prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
32	3	1	0	0	0	0	165,307,212	6.922%	4
49	2	1	0	0	0	0	123,534,096	5.173%	3
57	1	1	0	0	0	0	85,407,168	3.576%	2
59	0	1	0	0	0	0	47,618,592	1.994%	1
19	2	2	0	0	0	0	46,348,182	1.941%	4
25	4	1	0	0	0	0	32,134,536	1.346%	5
31	5	1	0	0	0	0	16,328,664	0.684%	6
17	3	2	0	0	0	0	12,903,645	0.540%	5
20	1	2	0	0	0	0	12,572,448	0.526%	3
56	2	0	1	0	0	0	11,412,702	0.478%	3
18	4	2	0	0	0	0	8,109,954	0.340%	6
23	3	1	1	0	0	0	4,840,596	0.203%	5
47	0	2	0	0	0	0	4,238,304	0.177%	2
62	3	0	1	0	0	0	3,186,837	0.133%	4
50	4	0	1	0	0	0	2,827,656	0.118%	5
21	3	3	0	0	0	0	1,700,664	0.071%	6
46	0	4	0	0	0	0	1,603,674	0.067%	4
44	1	1	1	0	0	0	1,480,608	0.062%	3
11	2	2	1	0	0	0	1,207,764	0.051%	5
37	6	1	0	0	0	0	1,068,480	0.045%	7
24	5	0	1	0	0	0	926,334	0.039%	6
48	0	3	0	0	0	0	899,304	0.038%	3
55	0	1	1	0	0	0	881,496	0.037%	2
13	0	2	1	0	0	0	845,244	0.035%	3
35	5	2	0	0	0	0	801,360	0.034%	7
43	2	1	1	0	0	0	369,198	0.015%	4
22	1	3	0	0	0	0	309,096	0.013%	4
60	4	0	0	1	0	0	294,840	0.012%	5
8	1	0	2	0	0	0	225,144	0.009%	3
70	2	3	1	0	0	0	213,696	0.009%	6
12	1	2	1	0	0	0	177,444	0.007%	4
16	2	3	0	0	0	0	148,824	0.006%	5
52	6	0	1	0	0	0	80,136	0.003%	7
72	3	4	0	0	0	0	71,232	0.003%	7
42	0	1	2	0	0	0	64,872	0.003%	3
71	5	0	2	0	0	0	53,424	0.002%	7
68	1	2	2	0	0	0	53,424	0.002%	5
66	3	2	0	1	0	0	52,416	0.002%	6
34	7	0	1	0	0	0	50,085	0.002%	8
53	1	1	0	1	0	0	46,800	0.002%	3
7	2	0	2	0	0	0	44,361	0.002%	4
5	0	1	1	1	0	0	34,632	0.001%	3
45	4	1	1	0	0	0	26,712	0.001%	6
65	1	4	1	0	0	0	26,712	0.001%	6
10	0	0	3	0	0	0	23,108	9.68 × 10-4 %	3
41	0	2	0	1	0	0	15,912	6.66 × 10-4 %	3
14	3	2	1	0	0	0	13,356	5.59 × 10-4 %	6
61	4	0	0	0	1	0	5,670	2.37 × 10-4 %	5
64	3	0	3	0	0	0	4,452	1.86 × 10-4 %	6
9	4	0	2	0	0	0	3,339	1.40 × 10-4 %	6
6	3	0	2	0	0	0	1,431	5.99 × 10-5 %	5
69	6	0	0	0	0	1	1,113	4.66 × 10-5 %	7
67	3	2	0	0	1	0	1,008	4.22 × 10-5 %	6
15	1	4	0	0	0	0	954	3.99 × 10-5 %	5
54	1	1	0	0	1	0	900	3.77 × 10-5 %	3
4	0	1	1	0	1	0	666	2.79 × 10-5 %	3
39	0	0	2	1	0	0	624	2.61 × 10-5 %	3
3	0	0	1	2	0	0	612	2.56 × 10-5 %	3
40	0	2	0	0	1	0	306	1.28 × 10-5 %	3
1	0	0	2	0	0	1	159	6.66 × 10-6 %	3
63	0	2	0	0	0	1	159	6.66 × 10-6 %	3
2	0	0	1	1	1	0	24	1.01 × 10-6 %	3
38	0	0	2	0	1	0	12	5.03 × 10-7 %	3
Total (out of 2,388,046,122) :							590,602,373	24.73%

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Match 3 Prizes

Outcomes of the 27 boards, awarding Match 3 prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
32	3	1	0	0	0	0	165,307,212	6.922%	4
49	2	1	0	0	0	0	123,534,096	5.173%	3
57	1	1	0	0	0	0	85,407,168	3.576%	2
59	0	1	0	0	0	0	47,618,592	1.994%	1
19	2	2	0	0	0	0	46,348,182	1.941%	4
25	4	1	0	0	0	0	32,134,536	1.346%	5
31	5	1	0	0	0	0	16,328,664	0.684%	6
17	3	2	0	0	0	0	12,903,645	0.540%	5
20	1	2	0	0	0	0	12,572,448	0.526%	3
18	4	2	0	0	0	0	8,109,954	0.340%	6
47	0	2	0	0	0	0	4,238,304	0.177%	2
21	3	3	0	0	0	0	1,700,664	0.071%	6
46	0	4	0	0	0	0	1,603,674	0.067%	4
37	6	1	0	0	0	0	1,068,480	0.045%	7
48	0	3	0	0	0	0	899,304	0.038%	3
35	5	2	0	0	0	0	801,360	0.034%	7
22	1	3	0	0	0	0	309,096	0.013%	4
16	2	3	0	0	0	0	148,824	0.006%	5
72	3	4	0	0	0	0	71,232	0.003%	7
15	1	4	0	0	0	0	954	3.99 × 10-5 %	5
Total (out of 2,388,046,122) :							561,106,389	23.50%

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Match 4 Prizes

Outcomes of the 27 boards, awarding Match 4 prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
56	2	0	1	0	0	0	11,412,702	0.478%	3
23	3	1	1	0	0	0	4,840,596	0.203%	5
62	3	0	1	0	0	0	3,186,837	0.133%	4
50	4	0	1	0	0	0	2,827,656	0.118%	5
44	1	1	1	0	0	0	1,480,608	0.062%	3
11	2	2	1	0	0	0	1,207,764	0.051%	5
24	5	0	1	0	0	0	926,334	0.039%	6
55	0	1	1	0	0	0	881,496	0.037%	2
13	0	2	1	0	0	0	845,244	0.035%	3
43	2	1	1	0	0	0	369,198	0.015%	4
8	1	0	2	0	0	0	225,144	0.009%	3
70	2	3	1	0	0	0	213,696	0.009%	6
12	1	2	1	0	0	0	177,444	0.007%	4
52	6	0	1	0	0	0	80,136	0.003%	7
42	0	1	2	0	0	0	64,872	0.003%	3
71	5	0	2	0	0	0	53,424	0.002%	7
68	1	2	2	0	0	0	53,424	0.002%	5
34	7	0	1	0	0	0	50,085	0.002%	8
7	2	0	2	0	0	0	44,361	0.002%	4
45	4	1	1	0	0	0	26,712	0.001%	6
65	1	4	1	0	0	0	26,712	0.001%	6
10	0	0	3	0	0	0	23,108	9.68 × 10-4 %	3
14	3	2	1	0	0	0	13,356	5.59 × 10-4 %	6
64	3	0	3	0	0	0	4,452	1.86 × 10-4 %	6
9	4	0	2	0	0	0	3,339	1.40 × 10-4 %	6
6	3	0	2	0	0	0	1,431	5.99 × 10-5 %	5
Total (out of 2,388,046,122) :							29,040,131	1.22%

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Match 5 Prizes

Outcomes of the 27 boards, awarding Match 5 prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
60	4	0	0	1	0	0	294,840	0.012%	5
66	3	2	0	1	0	0	52,416	0.002%	6
53	1	1	0	1	0	0	46,800	0.002%	3
5	0	1	1	1	0	0	34,632	0.001%	3
41	0	2	0	1	0	0	15,912	6.66 × 10-4 %	3
39	0	0	2	1	0	0	624	2.61 × 10-5 %	3
3	0	0	1	2	0	0	612	2.56 × 10-5 %	3
Total (out of 2,388,046,122) :							445,836	0.02%

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Match 5+ Prizes

Outcomes of the 27 boards, awarding Match 5+ prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
61	4	0	0	0	1	0	5,670	2.37 × 10-4 %	5
67	3	2	0	0	1	0	1,008	4.22 × 10-5 %	6
54	1	1	0	0	1	0	900	3.77 × 10-5 %	3
4	0	1	1	0	1	0	666	2.79 × 10-5 %	3
40	0	2	0	0	1	0	306	1.28 × 10-5 %	3
2	0	0	1	1	1	0	24	1.01 × 10-6 %	3
38	0	0	2	0	1	0	12	5.03 × 10-7 %	3
Total (out of 2,388,046,122) :							8,586	3.60 × 10-4 %

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Match 6 Prizes

Outcomes of the 27 boards, awarding Match 6 (jackpot) prizes

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	No. of Prizes
69	6	0	0	0	0	1	1,113	4.66 × 10-5 %	7
1	0	0	2	0	0	1	159	6.66 × 10-6 %	3
63	0	2	0	0	0	1	159	6.66 × 10-6 %	3
Total (out of 2,388,046,122) :							1,431	5.99 × 10-5 %

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Miscellaneous

Below is a table showing the 72 outcomes of playing the 27 boards proposed by the Manchester mathematicians. The list is in ascending order of ID. Also shown are the usual prizes, gains (or losses) and lists of six balls that would produce each outcome, for example.

Outcomes of the 27 boards, whichever balls are drawn, listed in ID order, showing usual prizes, gains/losses, and examples of balls drawn.

ID	Match 2	Match 3	Match 4	Match 5	Match 5+	Match 6	Frequency	Probability (%)	Usual Prize	Gain/Loss	Outcome Example
1	0	0	2	0	0	1	159	6.66 × 10-6 %	£2,000,280	£2,000,226	{1, 2, 3, 4, 5, 6}
2	0	0	1	1	1	0	24	1.01 × 10-6 %	£1,001,890	£1,001,836	{1, 2, 3, 4, 5, 7}
3	0	0	1	2	0	0	612	2.56 × 10-5 %	£3,640	£3,586	{1, 2, 3, 4, 5, 7}
4	0	1	1	0	1	0	666	2.79 × 10-5 %	£1,000,170	£1,000,116	{1, 2, 3, 4, 5, 9}
5	0	1	1	1	0	0	34,632	0.001%	£1,920	£1,866	{1, 2, 3, 4, 5, 9}
6	3	0	2	0	0	0	1,431	5.99 × 10-5 %	£280	£226	{1, 2, 3, 4, 9, 10}
7	2	0	2	0	0	0	44,361	0.002%	£280	£226	{1, 2, 3, 4, 9, 12}
8	1	0	2	0	0	0	225,144	0.009%	£280	£226	{1, 2, 3, 4, 9, 18}
9	4	0	2	0	0	0	3,339	1.40 × 10-4 %	£280	£226	{1, 2, 3, 4, 18, 19}
10	0	0	3	0	0	0	23,108	9.68 × 10-4 %	£420	£366	{1, 2, 3, 5, 7, 9}
11	2	2	1	0	0	0	1,207,764	0.051%	£200	£146	{1, 2, 3, 5, 9, 10}
12	1	2	1	0	0	0	177,444	0.007%	£200	£146	{1, 2, 3, 5, 9, 12}
13	0	2	1	0	0	0	845,244	0.035%	£200	£146	{1, 2, 3, 5, 9, 18}
14	3	2	1	0	0	0	13,356	5.59 × 10-4 %	£200	£146	{1, 2, 3, 5, 18, 19}
15	1	4	0	0	0	0	954	3.99 × 10-5 %	£120	£66	{1, 2, 3, 9, 10, 11}
16	2	3	0	0	0	0	148,824	0.006%	£90	£36	{1, 2, 3, 9, 10, 12}
17	3	2	0	0	0	0	12,903,645	0.540%	£60	£6	{1, 2, 3, 9, 10, 18}
18	4	2	0	0	0	0	8,109,954	0.340%	£60	£6	{1, 2, 3, 9, 12, 15}
19	2	2	0	0	0	0	46,348,182	1.941%	£60	£6	{1, 2, 3, 9, 12, 18}
20	1	2	0	0	0	0	12,572,448	0.526%	£60	£6	{1, 2, 3, 9, 18, 32}
21	3	3	0	0	0	0	1,700,664	0.071%	£90	£36	{1, 2, 3, 18, 19, 20}
22	1	3	0	0	0	0	309,096	0.013%	£90	£36	{1, 2, 3, 18, 20, 26}
23	3	1	1	0	0	0	4,840,596	0.203%	£170	£116	{1, 2, 9, 10, 11, 12}
24	5	0	1	0	0	0	926,334	0.039%	£140	£86	{1, 2, 9, 10, 12, 13}
25	4	1	0	0	0	0	32,134,536	1.346%	£30	-£24	{1, 2, 9, 10, 12, 18}
26	8	0	0	0	0	0	624,393	0.026%	£0	-£54	{1, 2, 9, 10, 18, 19}
27	6	0	0	0	0	0	48,824,289	2.045%	£0	-£54	{1, 2, 9, 10, 18, 20}
28	5	0	0	0	0	0	63,127,134	2.643%	£0	-£54	{1, 2, 9, 10, 18, 32}
29	7	0	0	0	0	0	5,011,839	0.210%	£0	-£54	{1, 2, 9, 12, 18, 19}
30	4	0	0	0	0	0	375,437,160	15.722%	£0	-£54	{1, 2, 9, 12, 18, 32}
31	5	1	0	0	0	0	16,328,664	0.684%	£30	-£24	{1, 2, 9, 18, 19, 20}
32	3	1	0	0	0	0	165,307,212	6.922%	£30	-£24	{1, 2, 9, 18, 20, 26}
33	3	0	0	0	0	0	457,275,308	19.149%	£0	-£54	{1, 2, 9, 18, 32, 46}
34	7	0	1	0	0	0	50,085	0.002%	£140	£86	{1, 2, 18, 19, 20, 21}
35	5	2	0	0	0	0	801,360	0.034%	£60	£6	{1, 2, 18, 19, 20, 22}
36	9	0	0	0	0	0	293,090	0.012%	£0	-£54	{1, 2, 18, 19, 32, 33}
37	6	1	0	0	0	0	1,068,480	0.045%	£30	-£24	{1, 2, 18, 20, 22, 26}
38	0	0	2	0	1	0	12	5.03 × 10-7 %	£1,000,280	£1,000,226	{1, 3, 4, 5, 6, 7}
39	0	0	2	1	0	0	624	2.61 × 10-5 %	£2,030	£1,976	{1, 3, 4, 5, 6, 7}
40	0	2	0	0	1	0	306	1.28 × 10-5 %	£1,000,060	£1,000,006	{1, 3, 4, 5, 6, 9}
41	0	2	0	1	0	0	15,912	6.66 × 10-4 %	£1,810	£1,756	{1, 3, 4, 5, 6, 9}
42	0	1	2	0	0	0	64,872	0.003%	£310	£256	{1, 3, 4, 5, 7, 9}
43	2	1	1	0	0	0	369,198	0.015%	£170	£116	{1, 3, 4, 5, 9, 12}
44	1	1	1	0	0	0	1,480,608	0.062%	£170	£116	{1, 3, 4, 5, 9, 18}
45	4	1	1	0	0	0	26,712	0.001%	£170	£116	{1, 3, 4, 5, 18, 19}
46	0	4	0	0	0	0	1,603,674	0.067%	£120	£66	{1, 3, 4, 9, 10, 11}
47	0	2	0	0	0	0	4,238,304	0.177%	£60	£6	{1, 3, 4, 9, 18, 32}
48	0	3	0	0	0	0	899,304	0.038%	£90	£36	{1, 3, 4, 18, 20, 26}
49	2	1	0	0	0	0	123,534,096	5.173%	£30	-£24	{1, 3, 5, 9, 18, 32}
50	4	0	1	0	0	0	2,827,656	0.118%	£140	£86	{1, 3, 9, 10, 12, 13}
51	2	0	0	0	0	0	611,250,696	25.596%	£0	-£54	{1, 3, 9, 18, 32, 46}
52	6	0	1	0	0	0	80,136	0.003%	£140	£86	{1, 3, 18, 19, 20, 21}
53	1	1	0	1	0	0	46,800	0.002%	£1,780	£1,726	{1, 9, 10, 11, 12, 13}
54	1	1	0	0	1	0	900	3.77 × 10-5 %	£1,000,030	£999,976	{1, 9, 10, 11, 12, 13}
55	0	1	1	0	0	0	881,496	0.037%	£170	£116	{1, 9, 10, 11, 12, 18}
56	2	0	1	0	0	0	11,412,702	0.478%	£140	£86	{1, 9, 10, 12, 13, 18}
57	1	1	0	0	0	0	85,407,168	3.576%	£30	-£24	{1, 9, 10, 12, 18, 32}
58	1	0	0	0	0	0	235,599,840	9.866%	£0	-£54	{1, 9, 12, 18, 32, 46}
59	0	1	0	0	0	0	47,618,592	1.994%	£30	-£24	{1, 9, 18, 20, 26, 32}
60	4	0	0	1	0	0	294,840	0.012%	£1,750	£1,696	{1, 18, 19, 20, 21, 26}
61	4	0	0	0	1	0	5,670	2.37 × 10-4 %	£1,000,000	£999,946	{1, 18, 19, 20, 21, 26}
62	3	0	1	0	0	0	3,186,837	0.133%	£140	£86	{3, 4, 5, 6, 9, 12}
63	0	2	0	0	0	1	159	6.66 × 10-6 %	£2,000,060	£2,000,006	{9, 10, 11, 12, 13, 14}
64	3	0	3	0	0	0	4,452	1.86 × 10-4 %	£420	£366	{18, 19, 20, 21, 22, 23}
65	1	4	1	0	0	0	26,712	0.001%	£260	£206	{18, 19, 20, 21, 22, 24}
66	3	2	0	1	0	0	52,416	0.002%	£1,810	£1,756	{18, 19, 20, 21, 22, 26}
67	3	2	0	0	1	0	1,008	4.22 × 10-5 %	£1,000,060	£1,000,006	{18, 19, 20, 21, 22, 26}
68	1	2	2	0	0	0	53,424	0.002%	£340	£286	{18, 19, 20, 21, 22, 28}
69	6	0	0	0	0	1	1,113	4.66 × 10-5 %	£2,000,000	£1,999,946	{18, 19, 20, 21, 26, 27}
70	2	3	1	0	0	0	213,696	0.009%	£230	£176	{18, 19, 20, 22, 24, 26}
71	5	0	2	0	0	0	53,424	0.002%	£280	£226	{18, 19, 20, 22, 26, 30}
72	3	4	0	0	0	0	71,232	0.003%	£120	£66	{18, 20, 22, 24, 26, 28}
Total (out of 2,388,046,122) :							2,388,046,122	100.00%

Revision date: 31 August 2023

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