Let’s face it. You are lucky to win any prize in Lotto. We can only do our best by maximising coverage.

You must match the six main balls drawn to win an exclusive jackpot of at least £2 million. Even so, it might be shared if there are other winners. Those are the rules.

Playing with 59 balls, you have one chance in 45,057,474, meaning the chances of winning are infinitesimal. However, other prizes in the hierarchy can be won more simply but have less value. Nevertheless, even they are more often than not beyond our reach.

Although buying many Lotto boards affords a greater but marginal opportunity for winning, there is the tantalising question of what you can do to maximise the outcome of your expenditure. The answers are nuanced.

Naturally, you want to win. However, the large, attractive prizes stem from the small losses made by most players, and there are millions of them. Lotteries work like this, and Lotto is no exception.

Let’s see what can be done in one approach that gets the best coverage by ensuring no triplet^[1] is repeated in the boards we play.

Estimated reading time: 0 minutes

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Lotto Prize Structure

The table below shows Lotto’s current prize structure.

Lotto Prizes

Category	Prize	Remarks
Match 2	Free Lucky Dip[2]	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[3]	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.

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Number Patterns

Of the 59 balls, 30 are odd numbers, and 29 are even.

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59} and

{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58}.

Likely composition of the six main balls drawn from 59

Composition	No. of Combinations	%	Example
6 odd numbers	593,775	1.3	1	11	23	29	33	57
5 odd and 1 even	4,132,674	9.2	3	9	18	21	31	55
4 odd and 2 even	11,126,430	24.7	3	5	11	32	46	49
3 odd and 3 even	14,835,240	32.9	1	8	16	23	25	56
2 odd and 4 even	10,331,685	22.9	3	14	17	20	38	54
1 odd and 5 even	3,562,650	7.9	1	12	20	26	52	48
6 even numbers	475,020	1.1	2	10	16	28	44	50
Total	45,057,474	100.0

There are no deep insights here. We need to look elsewhere to begin to get to grips with the problem of winning prizes in Lotto.

Whereas jackpot winners must match all six main numbers, the minimum cash prize is obtained by matching just three of the main numbers drawn by Camelot, which suggests a solution is possible by combining two minimum prizes somehow.

This approach is promising.

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Triplets

How many sets of three numbers can be chosen from 59? Indeed, there are many more options than the 18,424 available in the earlier version of Lotto that operated until Saturday, 10 October 2015.

There are ⁵⁹C₃ ways of choosing three numbers from 59.

$$ \begin{split}^{59}C_3& = \frac{59!}{3! × 56!}\\ \\&=\frac{(59×58×57×56×55×54×53×\dotso×6×5×4×3×2×1)}{(3×2×1)×(56×55×54×\dotso×3×2×1)}\\ \\&=32509 \end{split} $$

From now on, we shall refer to them as triplets for convenience and express them as a sequence of three numbers in ascending order, separated by commas and enclosed in braces (also known as curly brackets). e.g. {17, 39, 58}.

The number of triplets has increased by ~76% since the number of balls played rose from 49 to 59, making it more difficult to Match 3 balls than it used to be.

We can pair triplets to produce a sextet, corresponding to a draw of main numbers, provided that none of the numbers is repeated.

For example, pairing the first {1, 2, 3} and last triplet {57, 58, 59} produces the sextet: {1, 2, 3, 57, 58, 59} and pairing the last with the first has the same result. However, pairing triplets such as {1, 2, 3} and {1, 2, 59} makes no sense since balls 1 and 2 would be duplicated. Similarly, pairing any triplet with itself produces a sextet that would not be a valid Lotto board.

Test for inclusion

Using mathematical set theory, if A and B are sets built from triplets, only pairs for which \(A \cap B = \{\}\) produce valid sextets^[4].

For example, \(\{5, 17, 18\} \cap \{10, 30, 59\} = \{\}\) is valid, but \(\{1, 9, 46\} \cap \{40, 46, 59\} = \{46\}\) is not. The result is not the empty set \(\{\}\) but \(\{46\}\), since 46 is common to both triplets.

Constructing a square grid comprising 32,509 rows and 32,509 columns shows the pairing symmetry. The grid contains 1,056,835,081 cells.

A diagonal line (shown in grey) from the top left corner to the bottom right corner cuts the square into two equal triangles. It demonstrates that we need to consider only half of all possible pairings to determine how many form valid sextets, which reduces the effort considerably.

Nevertheless, despite the symmetry, even after discounting the 32,509 in the diagonal too, the number of pairs to be checked is still huge.

Taking away the number of cells in the diagonal (32,509) from the total (1,056,835,081) leaves 1,056,802,572. We need to check half this number (N). i.e. 528,401,286.

$$ \begin{align*}N& = \frac{32509 × 32508}{2}=528401286 \end{align*} $$

The checking exercise produces 491,080,954 valid sextets (92.9%) from the possible 528,401,286 pairs of triplets^[5].

Why are there 491,080,954 sextets instead of 45,057,474, as might be expected?

The apparent discrepancy arises because a sextet can be formed from more than one pair of triplets, meaning we have overcounted a lot. The reverse process demonstrates this by considering how many triplets a sextet produces.

Based on 59 balls, each triplet appears 27,720 times in the 45,057,474 Lotto boards.

$$ \begin{align*}frequency& = \frac{^{59}C_6 × ^{6}C_3}{^{59}C_3}\\ \\&=\frac{45057474 × 20}{32509}\\ \\&=27720 \end{align*} $$

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Triplets build sextets

Lotto boards are sextets – sets of six unique balls from 1 to 59.

There are ⁶C₃ ways of choosing three numbers from 6.

$$ \begin{split}^{6}C_3& = \frac{6!}{3! × 3!}\\ \\&=\frac{(6×5×4×3×2×1)}{(3×2×1)×(3×2×1)}\\ \\&=20 \end{split} $$

Consider the draw held on Saturday, 26 August 2023 – draw 2,888. The main numbers were: 12, 13, 41, 42, 45 and 50.

They form a sextet: \(\{12, 13, 41, 42, 45, 50\}\) from which we can obtain 20 triplets, shown in the table below.

Triplets formed from the sextet {12, 13, 41, 42, 45, 50}

{12, 13, 41}	{12, 13, 42}	{12, 13, 45}	{12, 13, 50}	{12, 41, 42}	{12, 41, 45}	{12, 41, 50}	{12, 42, 45}	{12, 42, 50}	{12, 45, 50}
{13, 41, 42}	{13, 41, 45}	{13, 41, 50}	{13, 42, 45}	{13, 42, 50}	{13, 45, 50}	{41, 42, 45}	{41, 42, 50}	{41, 45, 50}	{42, 45, 50}

Each one is a candidate for a Match 3 prize.

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Coverage

There are 20 chances of winning a Match 3 prize per board bought. Buying two boards should cover 40 of the basic 32,509 triplets.

 1, 15, 20, 21, 47, 51

 1,  6, 22, 29, 33, 46

However, buying the following two boards covers only 39 of the 32,509 triplets since the triplet {20, 21, 58} is duplicated.

  1, 15, 20, 21, 37, 58

10, 16, 20, 21, 41, 58

Drawing this triplet would win two Match 3 prizes; drawing any other triplet represented here would win only one Match 3 prize.

Consequently, the chances of winning a prize are enhanced significantly by the extent to which buying boards cover the 32,509 triplets. The degree of coverage is a valuable measure of likely success.

Any scheme that purports to win prizes in Lotto must be able at least to deliver Match 3 prizes reliably and consistently.

Using a suitable algorithm, I have prepared a list of 900 boards that cover 18,000 triplets without repetition for £1,800. However, the remaining 14,509 triplets can only be covered by repeating many triplets.

By buying the following boards, we could cover the available triplets fully in principle.

$$ \begin{align*}\frac{^{59}C_3}{^{6}C_3}=\frac{32509}{20}=1625.45 \; boards \end{align*} $$

We shall investigate how feasible this idea is in a later article.

The best I could do to cover all triplets took 2,183 boards, using 43,660 triplets, which means that 11,151 triplets were repeated, some more than twice. They produce a minimum guaranteed gain of twenty Match 3 prizes, whichever six balls are chosen from 59. A minimum cash prize of £600 for a stake of £4,366 is not such a bargain after all.

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900 Independent boards

My list follows. However, you can generate your Lotto boards randomly by clicking on the button. Pick as many boards as you like from the result to play Lotto. The sets you obtain are self-consistent in themselves. Good luck!

Choose your set of 900 boards

Independent boards from 59 balls with unique triplets

Board	Balls
1	1	55	56	57	58	59
2	1	51	52	53	54	59
3	1	48	49	50	51	58
4	1	46	47	50	52	57
5	1	45	46	49	53	56
6	1	45	47	48	54	55
7	1	43	44	50	53	55
8	1	42	43	49	54	57
9	1	42	44	48	52	56
10	1	41	42	47	53	58
11	1	41	43	46	48	59
12	1	41	44	45	51	57
13	1	40	41	50	54	56
14	1	40	42	46	51	55
15	1	40	43	45	52	58
16	1	40	44	47	49	59
17	1	39	40	48	53	57
18	1	39	41	49	52	55
19	1	39	42	45	50	59
20	1	39	43	47	51	56
21	1	39	44	46	54	58
22	1	35	36	37	38	59
23	1	32	33	34	35	58
24	1	30	31	34	36	57
25	1	29	30	33	37	56
26	1	29	31	32	38	55
27	1	27	28	34	37	55
28	1	26	27	33	36	54
29	1	26	28	31	35	56
30	1	25	26	34	38	53
31	1	25	27	30	32	59
32	1	25	28	29	36	58
33	1	24	25	31	33	52
34	1	24	26	32	37	57
35	1	24	27	29	35	53
36	1	24	28	30	38	54
37	1	22	23	33	38	57
38	1	21	22	32	36	56
39	1	21	23	31	37	58
40	1	20	21	30	35	55
41	1	20	22	29	34	59
42	1	20	23	28	32	53
43	1	19	20	27	31	51
44	1	19	21	28	33	59
45	1	19	22	26	30	58
46	1	19	23	25	35	54
47	1	18	19	24	34	56
48	1	18	20	25	37	50
49	1	18	21	27	38	52
50	1	18	23	26	29	51
51	1	17	18	22	28	49
52	1	17	20	24	36	48
53	1	16	17	23	27	56
54	1	15	16	22	24	55
55	1	15	17	21	25	57
56	1	14	15	23	30	52
57	1	14	16	21	26	50
58	1	14	17	19	29	47
59	1	13	14	22	25	51
60	1	13	15	20	26	49
61	1	13	16	19	32	52
62	1	12	13	23	24	59
63	1	12	14	20	33	46
64	1	12	15	19	36	53
65	1	12	16	18	30	48
66	1	11	12	22	27	50
67	1	11	13	21	29	54
68	1	11	14	18	31	59
69	1	11	16	20	38	58
70	1	10	11	23	34	49
71	1	10	12	17	26	55
72	1	10	13	18	33	53
73	1	9	10	22	31	54
74	1	9	11	19	37	48
75	1	9	12	21	34	51
76	1	9	13	17	30	50
77	1	9	15	18	32	47
78	1	8	9	23	36	55
79	1	8	10	21	24	47
80	1	8	11	17	32	51
81	1	7	8	22	35	52
82	1	7	9	16	25	49
83	1	7	10	19	38	50
84	1	7	11	15	28	46
85	1	6	7	18	36	45
86	1	6	8	16	28	57
87	1	6	9	14	24	58
88	1	6	10	15	27	48
89	1	5	6	22	37	53
90	1	5	7	17	31	44
91	1	5	8	15	29	50
92	1	5	10	16	35	51
93	1	4	5	14	27	57
94	1	4	6	17	33	43
95	1	4	7	13	34	48
96	1	4	8	12	25	56
97	1	3	4	18	35	46
98	1	3	5	13	28	47
99	1	3	6	12	29	52
100	1	3	7	14	32	54
101	1	2	3	17	34	45
102	1	2	4	16	29	44
103	1	2	5	12	32	49
104	1	2	6	13	31	46
105	1	2	8	14	37	43
106	2	50	51	54	55	57
107	2	50	52	53	56	58
108	2	48	49	54	56	59
109	2	46	47	53	55	59
110	2	45	46	48	51	52
111	2	45	47	49	57	58
112	2	43	44	52	57	59
113	2	42	43	48	55	58
114	2	42	44	49	51	53
115	2	41	42	46	56	57
116	2	41	43	45	53	54
117	2	41	44	47	48	50
118	2	40	41	51	58	59
119	2	40	42	47	52	54
120	2	40	43	46	49	50
121	2	40	44	45	55	56
122	2	34	35	38	39	57
123	2	34	36	37	40	53
124	2	33	35	37	41	55
125	2	33	36	38	42	50
126	2	32	35	36	43	51
127	2	32	37	38	44	58
128	2	31	32	34	41	52
129	2	30	31	38	40	48
130	2	30	32	33	39	54
131	2	29	30	36	41	49
132	2	29	31	37	39	59
133	2	29	33	34	43	47
134	2	28	29	38	51	56
135	2	28	30	37	42	45
136	2	28	31	36	44	54
137	2	27	28	35	46	58
138	2	27	29	32	40	57
139	2	26	27	38	45	59
140	2	26	28	33	48	57
141	2	26	30	34	44	46
142	2	25	26	37	47	56
143	2	25	27	36	39	55
144	2	25	28	34	50	59
145	2	24	25	32	48	53
146	2	24	26	35	49	52
147	2	24	27	31	43	56
148	2	23	24	37	46	54
149	2	23	27	30	47	51
150	2	23	29	35	45	50
151	2	22	23	32	42	59
152	2	22	24	28	39	47
153	2	21	22	38	49	55
154	2	21	23	26	39	53
155	2	21	25	31	35	42
156	2	20	21	34	54	58
157	2	20	22	31	33	53
158	2	20	23	25	38	52
159	2	19	20	26	29	55
160	2	19	21	27	37	50
161	2	18	19	30	35	59
162	2	18	21	24	33	51
163	2	18	22	27	34	48
164	2	17	18	26	31	58
165	2	17	19	25	33	49
166	2	16	17	30	52	55
167	2	16	18	23	28	43
168	2	16	19	24	36	58
169	2	15	16	21	32	46
170	2	15	17	24	29	42
171	2	15	18	20	36	57
172	2	15	22	25	30	43
173	2	14	15	27	33	44
174	2	14	16	22	35	56
175	2	14	17	21	28	41
176	2	14	18	25	29	54
177	2	14	19	23	31	57
178	2	14	20	24	30	50
179	2	13	15	23	34	56
180	2	13	16	20	27	42
181	2	13	17	22	37	57
182	2	12	13	21	36	52
183	2	12	17	20	35	47
184	2	11	12	30	53	57
185	2	11	13	28	32	55
186	2	11	15	19	38	54
187	2	10	11	22	26	51
188	2	10	12	24	38	41
189	2	10	13	19	39	48
190	2	9	10	23	33	58
191	2	9	11	20	39	56
192	2	9	12	22	29	46
193	2	9	13	14	26	40
194	2	8	9	19	28	53
195	2	8	10	20	32	45
196	2	8	11	16	25	41
197	2	8	12	18	39	50
198	2	7	8	24	34	55
199	2	7	9	18	37	52
200	2	7	10	21	30	56
201	2	7	11	23	36	48
202	2	7	12	16	26	54
203	2	6	7	20	28	49
204	2	6	8	21	29	48
205	2	6	9	17	27	54
206	2	6	10	14	34	42
207	2	6	12	15	37	51
208	2	5	6	18	38	47
209	2	5	7	19	22	45
210	2	5	8	17	23	40
211	2	5	9	16	31	50
212	2	5	10	15	28	52
213	2	4	5	20	37	48
214	2	4	6	26	32	50
215	2	4	7	17	38	53
216	2	4	8	15	31	49
217	2	3	4	14	36	59
218	2	3	5	11	21	44
219	2	3	6	16	33	40
220	2	3	7	15	39	58
221	2	3	8	13	35	54
222	3	49	50	53	57	59
223	3	48	51	53	55	56
224	3	48	52	54	57	58
225	3	46	47	49	51	54
226	3	45	46	50	55	58
227	3	45	47	52	56	59
228	3	43	44	49	56	58
229	3	42	43	50	51	52
230	3	42	44	54	55	59
231	3	41	42	45	48	49
232	3	41	43	47	55	57
233	3	41	44	46	52	53
234	3	38	39	40	42	56
235	3	36	37	39	41	54
236	3	35	36	40	44	57
237	3	34	35	37	42	47
238	3	34	36	38	43	48
239	3	33	34	39	44	51
240	3	33	35	38	45	53
241	3	32	33	37	43	59
242	3	32	34	40	49	55
243	3	32	35	39	48	50
244	3	31	32	36	42	58
245	3	31	37	40	45	51
246	3	30	31	39	43	53
247	3	30	32	38	41	51
248	3	29	30	40	46	59
249	3	29	31	34	50	56
250	3	28	29	39	45	57
251	3	28	30	33	36	55
252	3	28	31	38	49	52
253	3	27	28	40	43	54
254	3	27	29	38	44	47
255	3	27	31	35	41	59
256	3	26	27	37	46	48
257	3	26	28	34	41	58
258	3	25	26	32	44	45
259	3	25	27	33	42	57
260	3	24	25	40	47	58
261	3	24	27	30	34	52
262	3	24	28	37	44	50
263	3	23	24	33	41	56
264	3	23	25	30	37	49
265	3	23	26	38	50	54
266	3	22	23	36	47	53
267	3	22	24	31	46	57
268	3	22	25	29	35	55
269	3	21	22	30	45	54
270	3	21	23	28	35	51
271	3	21	24	26	29	43
272	3	21	31	33	47	48
273	3	20	21	37	38	57
274	3	20	22	27	39	49
275	3	20	25	36	46	56
276	3	19	20	30	44	48
277	3	19	21	25	39	52
278	3	19	29	37	53	58
279	3	18	19	26	36	49
280	3	18	20	24	42	53
281	3	18	22	38	58	59
282	3	17	18	32	56	57
283	3	17	19	28	42	46
284	3	17	20	29	33	54
285	3	17	22	26	40	52
286	3	16	17	25	48	59
287	3	16	19	22	41	50
288	3	16	20	23	31	55
289	3	15	16	34	53	54
290	3	15	17	27	36	51
291	3	15	18	23	40	48
292	3	14	15	20	35	43
293	3	14	16	18	27	45
294	3	14	17	24	38	55
295	3	13	14	23	29	42
296	3	13	15	37	52	55
297	3	13	16	26	30	57
298	3	13	21	27	32	53
299	3	12	13	20	40	50
300	3	12	14	26	39	47
301	3	12	15	22	28	56
302	3	12	19	23	43	45
303	3	11	12	25	31	54
304	3	11	13	24	36	45
305	3	11	15	30	47	50
306	3	10	11	20	28	59
307	3	10	12	30	35	58
308	3	10	15	19	24	32
309	3	9	10	21	36	50
310	3	9	11	23	32	52
311	3	9	13	19	34	59
312	3	8	9	20	26	51
313	3	8	10	23	39	46
314	3	8	11	19	27	56
315	3	8	12	17	21	49
316	3	7	8	18	25	43
317	3	7	9	12	24	48
318	3	7	10	13	17	41
319	3	7	11	16	29	51
320	3	6	7	23	34	57
321	3	6	8	14	22	44
322	3	6	9	18	39	55
323	3	5	6	24	51	59
324	3	5	8	16	36	52
325	3	5	9	15	25	41
326	3	5	10	14	33	49
327	3	4	5	7	26	56
328	3	4	6	25	28	53
329	4	49	51	52	56	57
330	4	49	53	54	55	58
331	4	48	50	52	55	59
332	4	46	47	48	56	58
333	4	45	46	54	57	59
334	4	45	47	50	51	53
335	4	43	44	48	51	54
336	4	42	43	53	56	59
337	4	42	44	50	57	58
338	4	38	39	44	45	52
339	4	37	38	43	46	55
340	4	36	38	40	41	47
341	4	36	39	42	46	49
342	4	35	37	39	40	58
343	4	34	35	43	45	49
344	4	34	37	41	44	59
345	4	33	34	40	46	52
346	4	32	33	44	47	55
347	4	32	34	38	42	54
348	4	32	35	41	53	57
349	4	31	33	41	42	51
350	4	30	31	37	47	54
351	4	30	32	36	45	48
352	4	30	33	38	49	59
353	4	30	34	39	55	56
354	4	29	30	35	42	52
355	4	29	39	41	43	50
356	4	28	29	40	48	49
357	4	28	32	39	51	59
358	4	28	33	35	50	54
359	4	27	28	41	45	56
360	4	27	29	34	51	58
361	4	27	30	40	44	53
362	4	26	27	43	47	52
363	4	26	29	31	36	53
364	4	25	26	40	55	57
365	4	25	27	35	38	48
366	4	24	25	36	37	51
367	4	24	26	33	45	58
368	4	23	24	32	40	43
369	4	23	25	31	44	46
370	4	22	23	41	52	58
371	4	22	24	38	50	56
372	4	22	28	36	43	57
373	4	21	22	35	47	59
374	4	21	23	42	45	55
375	4	21	24	30	41	46
376	4	20	21	31	40	56
377	4	20	23	39	47	57
378	4	19	20	36	52	54
379	4	19	21	26	38	51
380	4	19	22	32	37	49
381	4	19	23	33	48	53
382	4	19	24	31	35	55
383	4	18	19	25	42	47
384	4	18	20	30	43	58
385	4	18	22	26	39	54
386	4	18	23	34	36	50
387	4	17	18	29	37	45
388	4	17	20	27	32	46
389	4	17	23	26	28	30
390	4	16	17	24	34	57
391	4	16	20	22	25	45
392	4	16	21	37	52	53
393	4	15	16	33	36	56
394	4	15	17	19	40	59
395	4	15	18	21	28	44
396	4	14	15	26	37	42
397	4	14	16	31	39	48
398	4	14	21	25	32	58
399	4	13	14	24	28	52
400	4	13	15	29	38	57
401	4	13	16	18	40	51
402	4	13	17	35	44	56
403	4	12	13	27	33	39
404	4	12	14	22	34	53
405	4	12	15	20	41	55
406	4	11	12	26	44	49
407	4	11	13	19	46	50
408	4	11	14	23	54	56
409	4	11	15	24	39	53
410	4	11	17	22	42	48
411	4	10	11	33	37	57
412	4	10	12	18	31	52
413	4	10	16	27	49	50
414	4	9	10	29	32	56
415	4	9	11	28	31	58
416	4	9	12	16	23	38
417	4	9	14	18	41	49
418	4	8	9	24	27	59
419	4	8	10	26	41	48
420	4	7	8	19	30	57
421	4	7	9	22	51	55
422	4	7	10	20	24	44
423	4	6	7	16	58	59
424	4	6	8	13	23	37
425	4	6	11	21	27	36
426	4	5	6	12	40	54
427	4	5	8	11	18	55
428	4	5	9	17	21	39
429	4	5	10	13	22	30
430	5	47	48	49	52	53
431	5	47	50	54	58	59
432	5	46	51	53	57	58
433	5	46	52	54	55	56
434	5	45	48	50	56	57
435	5	43	44	45	46	47
436	5	39	40	43	55	59
437	5	38	39	41	48	58
438	5	38	40	44	50	51
439	5	37	38	45	49	54
440	5	37	39	42	52	57
441	5	36	39	44	53	56
442	5	35	36	49	50	55
443	5	34	36	42	45	51
444	5	33	34	41	50	53
445	5	33	35	44	48	59
446	5	32	35	40	47	56
447	5	31	32	45	53	59
448	5	31	33	36	37	46
449	5	31	40	41	49	57
450	5	30	32	42	46	50
451	5	29	30	44	54	57
452	5	29	31	35	43	58
453	5	29	32	33	51	52
454	5	28	29	42	53	55
455	5	28	30	43	49	51
456	5	27	28	32	36	38
457	5	27	30	35	39	45
458	5	27	31	42	48	54
459	5	27	34	46	49	59
460	5	26	27	44	55	58
461	5	26	30	41	52	59
462	5	25	26	31	39	51
463	5	25	28	33	40	45
464	5	25	32	34	43	57
465	5	24	25	42	49	56
466	5	24	26	34	40	48
467	5	24	27	37	41	47
468	5	23	26	33	42	47
469	5	23	28	37	56	59
470	5	22	24	33	43	54
471	5	22	26	35	38	46
472	5	22	31	34	47	55
473	5	21	22	40	42	58
474	5	21	23	36	41	43
475	5	21	25	29	38	59
476	5	20	21	24	45	52
477	5	20	22	41	51	56
478	5	19	20	38	42	43
479	5	19	21	32	48	55
480	5	19	23	34	44	52
481	5	19	24	29	39	46
482	5	18	19	28	41	54
483	5	18	21	30	34	37
484	5	18	23	25	27	53
485	5	17	25	30	36	47
486	5	17	26	37	43	50
487	5	16	18	24	32	44
488	5	16	20	28	34	39
489	5	16	22	23	29	49
490	5	15	16	19	47	57
491	5	15	23	24	31	38
492	5	14	15	22	32	39
493	5	14	16	30	38	53
494	5	14	17	34	35	54
495	5	14	20	29	36	40
496	5	13	14	41	45	55
497	5	13	15	18	35	42
498	5	13	16	25	37	58
499	5	12	13	19	31	56
500	5	12	14	25	50	52
501	5	12	15	17	33	58
502	5	12	24	28	35	57
503	5	11	12	16	42	59
504	5	11	13	27	40	52
505	5	11	20	26	32	54
506	5	10	11	25	46	48
507	5	9	10	38	55	57
508	5	9	11	14	47	51
509	5	9	12	18	26	45
510	5	9	13	20	23	46
511	5	8	9	34	56	58
512	5	8	10	12	27	43
513	5	8	13	21	33	57
514	5	7	8	14	28	48
515	5	7	11	24	30	58
516	5	7	15	21	51	54
517	5	6	7	13	39	50
518	5	6	8	20	25	35
519	5	6	9	19	30	33
520	5	6	11	17	29	56
521	6	47	51	52	55	58
522	6	47	53	54	56	57
523	6	46	48	49	55	57
524	6	44	45	48	53	58
525	6	44	46	50	56	59
526	6	40	41	42	43	44
527	6	37	38	41	50	52
528	6	36	37	43	47	49
529	6	35	36	41	48	56
530	6	35	38	42	49	58
531	6	35	39	43	46	52
532	6	34	39	47	48	59
533	6	33	36	39	57	58
534	6	33	44	49	52	54
535	6	32	33	42	45	56
536	6	32	36	40	52	59
537	6	31	34	35	51	53
538	6	31	38	43	45	57
539	6	30	34	43	50	54
540	6	29	32	41	46	58
541	6	29	34	36	44	55
542	6	29	39	42	51	54
543	6	28	30	40	56	58
544	6	28	31	42	47	50
545	6	28	32	37	48	54
546	6	28	35	45	55	59
547	6	27	31	32	39	44
548	6	27	33	38	46	51
549	6	26	27	41	49	53
550	6	26	29	40	45	47
551	6	25	26	30	42	48
552	6	25	34	37	45	46
553	6	24	25	41	54	55
554	6	24	26	28	36	46
555	6	24	30	35	44	47
556	6	23	24	39	45	49
557	6	23	26	43	55	56
558	6	22	23	27	35	40
559	6	22	25	31	49	59
560	6	21	22	28	34	52
561	6	21	23	25	33	50
562	6	21	24	38	40	53
563	6	20	22	30	32	57
564	6	19	20	23	41	59
565	6	19	25	27	29	43
566	6	18	19	40	50	57
567	6	18	20	26	33	34
568	6	18	27	37	42	59
569	6	17	18	30	46	53
570	6	17	19	31	48	52
571	6	17	20	37	40	55
572	6	16	17	26	38	44
573	6	16	18	25	52	56
574	6	16	20	36	50	53
575	6	15	16	30	31	41
576	6	15	17	22	45	50
577	6	15	19	21	49	56
578	6	15	29	33	53	59
579	6	14	15	25	36	38
580	6	14	16	32	49	51
581	6	14	29	35	37	57
582	6	13	14	21	30	59
583	6	13	16	24	43	48
584	6	13	19	28	44	51
585	6	13	22	33	41	47
586	6	12	13	17	25	32
587	6	12	19	22	24	42
588	6	11	12	28	38	39
589	6	11	14	19	53	55
590	6	11	15	26	52	57
591	6	11	16	23	46	47
592	6	10	11	18	32	43
593	6	10	12	21	44	57
594	6	10	13	26	54	58
595	6	10	23	29	30	38
596	6	9	10	25	40	51
597	6	9	13	42	52	53
598	6	8	9	11	49	50
599	6	8	10	31	33	55
600	6	7	9	21	26	35
601	6	7	12	27	30	55
602	7	46	48	50	53	54
603	7	45	52	53	55	57
604	7	42	47	51	57	59
605	7	39	40	41	45	46
606	7	37	39	44	48	55
607	7	37	40	43	56	57
608	7	36	38	54	56	58
609	7	34	41	47	49	56
610	7	33	35	40	49	51
611	7	32	34	37	50	51
612	7	32	38	39	43	49
613	7	31	33	34	45	54
614	7	31	35	36	39	47
615	7	30	33	43	48	52
616	7	30	44	45	49	50
617	7	29	30	32	47	53
618	7	29	33	42	44	46
619	7	28	31	37	41	53
620	7	27	28	29	50	52
621	7	27	34	39	42	53
622	7	26	28	38	47	55
623	7	26	31	40	50	59
624	7	26	36	44	51	52
625	7	25	29	41	48	57
626	7	25	33	39	56	59
627	7	24	27	36	49	57
628	7	24	32	35	46	59
629	7	23	25	45	51	58
630	7	23	27	41	44	54
631	7	22	25	37	38	42
632	7	22	28	32	40	44
633	7	22	34	43	46	58
634	7	21	23	49	52	59
635	7	21	25	27	46	47
636	7	20	21	41	42	50
637	7	20	25	30	34	40
638	7	20	27	33	37	58
639	7	19	20	35	53	56
640	7	19	24	41	43	51
641	7	18	19	23	46	55
642	7	18	20	38	48	51
643	7	18	21	22	29	31
644	7	17	18	33	47	50
645	7	17	19	21	34	36
646	7	17	20	43	45	59
647	7	17	24	39	52	54
648	7	16	19	40	42	48
649	7	16	32	36	41	55
650	7	15	16	35	37	45
651	7	15	17	26	32	48
652	7	15	31	42	55	56
653	7	14	16	20	47	52
654	7	14	17	46	51	56
655	7	14	18	40	53	58
656	7	14	22	23	24	26
657	7	13	14	27	31	38
658	7	13	15	25	44	53
659	7	12	13	37	46	49
660	7	12	14	36	42	43
661	7	12	15	38	40	52
662	7	12	18	34	35	44
663	7	11	12	21	32	45
664	7	11	35	43	54	55
665	7	10	14	29	55	59
666	7	10	15	23	43	47
667	7	10	16	22	39	57
668	7	9	11	38	41	59
669	7	9	13	32	57	58
670	7	8	9	29	40	54
671	7	8	11	26	33	53
672	8	42	46	52	58	59
673	8	37	38	39	47	53
674	8	37	49	51	55	59
675	8	36	41	46	50	51
676	8	35	40	42	50	53
677	8	32	38	48	57	59
678	8	31	32	43	46	54
679	8	31	38	41	44	56
680	8	30	39	40	51	52
681	8	29	30	43	45	55
682	8	29	32	35	44	49
683	8	29	36	42	47	56
684	8	28	33	39	42	43
685	8	27	30	38	50	58
686	8	27	36	37	44	45
687	8	26	28	45	52	54
688	8	26	32	34	36	39
689	8	25	34	44	47	54
690	8	24	35	45	51	56
691	8	23	24	44	53	57
692	8	23	30	34	35	48
693	8	22	24	37	48	58
694	8	22	27	41	42	55
695	8	22	34	40	45	57
696	8	21	22	43	51	53
697	8	21	28	50	55	56
698	8	21	32	37	40	41
699	8	20	21	39	44	59
700	8	20	24	38	46	49
701	8	20	27	40	47	48
702	8	20	29	31	52	57
703	8	19	23	32	47	58
704	8	19	24	25	26	50
705	8	18	19	29	33	38
706	8	18	21	35	36	58
707	8	18	24	28	31	40
708	8	17	19	35	39	41
709	8	17	22	36	54	59
710	8	16	17	33	45	46
711	8	16	18	34	42	49
712	8	16	20	37	54	56
713	8	16	26	35	43	59
714	8	15	16	44	51	58
715	8	15	17	43	52	56
716	8	15	18	41	45	59
717	8	15	27	39	54	57
718	8	14	15	19	34	46
719	8	14	17	30	31	42
720	8	14	33	41	54	58
721	8	13	14	39	49	56
722	8	13	17	29	34	53
723	8	13	18	44	46	48
724	8	13	19	36	40	43
725	8	13	20	28	30	41
726	8	12	13	38	42	51
727	8	12	14	16	24	29
728	8	11	12	34	37	52
729	8	11	15	20	22	23
730	8	10	11	14	35	38
731	8	10	17	25	28	37
732	8	9	13	31	45	47
733	8	9	22	25	32	33
734	9	35	37	44	46	51
735	9	35	38	43	47	50
736	9	33	36	41	45	52
737	9	31	34	44	48	49
738	9	31	36	51	56	59
739	9	30	35	36	53	54
740	9	30	42	47	49	55
741	9	28	30	39	46	48
742	9	27	29	33	48	55
743	9	26	27	42	50	56
744	9	26	28	29	44	59
745	9	26	31	41	46	55
746	9	26	33	37	39	49
747	9	25	28	43	52	55
748	9	25	30	38	45	56
749	9	24	28	32	34	45
750	9	23	25	29	34	39
751	9	23	28	41	50	57
752	9	21	23	44	47	56
753	9	21	27	40	45	49
754	9	20	22	44	50	52
755	9	20	24	35	40	41
756	9	20	25	53	58	59
757	9	19	21	43	46	57
758	9	19	22	36	38	39
759	9	19	42	45	54	58
760	9	18	31	33	35	57
761	9	17	18	36	40	42
762	9	17	23	43	49	53
763	9	17	29	38	52	58
764	9	16	18	21	54	59
765	9	16	22	37	40	47
766	9	16	27	41	43	58
767	9	15	19	27	35	52
768	9	15	20	28	33	38
769	9	15	24	36	43	44
770	9	15	30	37	57	59
771	9	14	15	21	31	53
772	9	14	20	42	48	57
773	9	13	18	22	43	56
774	9	13	24	29	49	51
775	9	12	14	37	54	55
776	9	12	19	32	41	44
777	9	12	33	40	53	56
778	9	11	13	16	33	44
779	9	11	15	17	34	55
780	9	10	12	15	39	42
781	9	10	14	17	44	45
782	9	10	16	26	34	52
783	10	37	42	51	56	58
784	10	35	37	43	48	53
785	10	31	38	42	46	53
786	10	29	35	41	47	54
787	10	29	37	40	44	52
788	10	28	36	47	48	51
789	10	26	28	32	42	49
790	10	26	33	38	40	43
791	10	25	30	39	41	44
792	10	24	29	31	45	48
793	10	24	30	40	42	57
794	10	24	33	34	36	59
795	10	23	26	36	45	57
796	10	23	40	41	53	55
797	10	22	27	36	46	52
798	10	21	28	38	45	58
799	10	20	21	29	46	51
800	10	20	27	34	38	56
801	10	20	31	39	50	58
802	10	19	22	44	53	59
803	10	19	25	34	55	58
804	10	19	27	33	45	47
805	10	18	20	40	49	54
806	10	18	28	34	46	57
807	10	17	18	23	24	35
808	10	17	19	30	51	54
809	10	17	21	33	42	52
810	10	17	22	32	38	47
811	10	16	17	31	36	43
812	10	16	24	28	53	56
813	10	15	21	34	35	40
814	10	14	15	18	50	56
815	10	14	16	37	41	46
816	10	14	21	39	43	54
817	10	13	23	32	44	50
818	10	13	25	45	49	52
819	10	12	14	32	40	48
820	10	11	15	36	41	58
821	11	30	31	44	51	55
822	11	29	33	40	50	58
823	11	26	31	34	42	43
824	11	25	28	44	56	57
825	11	24	27	28	42	51
826	11	24	31	32	47	49
827	11	23	27	39	58	59
828	11	22	24	29	41	44
829	11	22	33	39	45	55
830	11	21	33	35	46	56
831	11	20	21	43	48	49
832	11	20	35	42	44	45
833	11	19	21	23	30	40
834	11	19	28	29	34	35
835	11	18	21	39	41	57
836	11	18	26	35	47	48
837	11	18	28	36	52	53
838	11	17	21	31	38	50
839	11	17	27	35	37	49
840	11	16	32	34	48	56
841	11	15	18	25	49	51
842	11	14	17	43	57	58
843	11	14	22	30	36	37
844	11	14	29	46	49	52
845	11	13	20	37	51	53
846	11	13	25	43	47	59
847	11	12	17	24	40	46
848	12	29	32	36	50	54
849	12	27	28	48	53	59
850	12	26	30	33	50	51
851	12	26	32	43	53	58
852	12	25	35	39	49	53
853	12	23	28	34	40	47
854	12	23	37	42	48	50
855	12	22	25	36	44	48
856	12	21	24	39	55	58
857	12	20	21	25	26	28
858	12	20	23	36	49	58
859	12	20	31	37	43	44
860	12	19	25	46	51	59
861	12	19	26	27	34	57
862	12	18	27	51	54	56
863	12	18	29	43	49	59
864	12	17	22	23	31	51
865	12	16	19	28	49	55
866	12	15	21	23	27	29
867	12	15	24	34	49	50
868	12	15	31	35	46	48
869	13	25	38	40	46	54
870	13	24	26	39	41	42
871	13	22	23	28	45	48
872	13	20	24	47	55	56
873	13	18	20	29	32	39
874	13	18	25	31	36	41
875	13	16	21	35	38	41
876	13	15	17	39	46	47
877	13	15	21	43	50	58
878	13	15	31	32	33	40
879	13	14	34	36	47	58
880	14	27	32	37	52	56
881	14	23	32	38	45	46
882	14	23	34	43	51	59
883	14	20	22	28	55	58
884	14	19	20	37	39	45
885	14	19	24	40	44	54
886	14	19	28	32	43	50
887	14	18	26	30	32	55
888	14	17	27	39	40	50
889	14	17	32	33	36	53
890	14	15	28	40	51	57
891	15	29	36	39	48	52
892	15	21	26	36	55	59
893	15	20	29	45	56	58
894	15	19	23	39	50	51
895	16	24	38	42	45	47
896	16	23	30	36	42	44
897	16	21	27	48	51	57
898	16	19	21	31	44	45
899	16	17	32	40	54	58
900	17	21	30	32	43	44

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Notes

• [1] A triplet is a set of three numbers written in ascending order, separated by commas and enclosed in braces. e.g. {1, 2, 3} and {17, 30, 59}. Under normal circumstances, Lotto’s minimum cash prize of £30 is awarded for matching three numbers corresponding to a triplet. Triplets are interesting since two dissimilar triplets comprise each Lotto board (sextet).
• [2] 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.
• [3] Usually, Lotto’s minimum cash prize is £30.
• [4] In the language of set theory, the symbol ∩ is a mathematical operator that finds the intersection of two sets. i.e. their common elements. Note that {} is the empty set, implying nothing in common.
• [5] This value can be found by calculating the square of 32509 and subtracting 32509 from the result to account for the shaded diagonal that we do not need to include. i.e. 32509² – 32509, equivalent to 32509 × (32509 – 1). Then, divide by 2.
  It is relatively easy to write a computer program in Pascal/Delphi to accomplish this task, taking advantage of the mathematical properties of sets that the languages provide.

Revision date: 29 August 2023