Ask a Language Model for Lottery Numbers: Concentration in Repeated Six-of-49 Outputs

Dmitrij Żatuchin

cs.IRSep 4, 2026

Ask a Language Model for Lottery Numbers: Concentration in Repeated Six-of-49 Outputs

Authors: Dmitrij Żatuchin

Organizations: Department of Information Technologies, Estonian Entrepreneurship University of Applied Sciences (EUAS), Tallinn, Estonia · Rankfor.AI, Tallinn, Estonia

Abstract

We evaluate six language-model configurations on requests for six distinct random integers from 1-49. Across 1,200 attempted calls using four English prompt variants, 1,184 responses yielded valid tickets. Effective diversity of number frequencies ranged from 9.9 to 18.0, compared with simulated fifth-percentile thresholds of 46.6-46.7 under independent uniform six-of-49 sampling at the corresponding sample sizes. Systems produced 8-93 distinct unordered tickets, and their modal tickets accounted for 22.5-68.0% of valid responses. Two archived Polish Lotto samples provided a physical-lottery comparison, with effective diversities of 41.1 and 41.4 at smaller sample sizes. These results demonstrate substantial concentration under the tested deployment settings. They do not identify its mechanism or establish performance under other prompts, temperatures, or tool configurations.

Figures & tables

Figure 1: Share of number occurrences, 1 to 49, on a shared y-scale. Top row: the two archived Polish Lotto samples, 26 and 39 draws, marked as smaller archival samples. Rows two to four: six engines, 188 to 200 valid tickets each, four English prompt variants, provider-default settings, fresh context per call, 1 September 2026. Mint bars mark the numbers in each engine’s modal unordered ticket. The dashed line is the uniform share $100/49=2.04\%$ . Engine panels are ordered by descending effective diversity; the order is presentational and carries no quality ranking.

source	tickets	numbers	$N_{\mathrm{eff}}$ of 49	floor	$p_{\mathrm{MC}}$	top count (p95)
Polish Lotto, summer 1999	26	156	41.1	35.7	0.95	7 (9)
Polish Lotto, summer 2026	39	234	41.4	39.2	0.49	10 (12)
grok-4.5	200	1,200	18.0	46.7	0.00005	115 (40)
claude-sonnet-5	196	1,176	13.5	46.7	0.00005	169 (39)
mistral-large	188	1,128	13.2	46.6	0.00005	161 (38)
gpt-5.6-luna	200	1,200	12.4	46.7	0.00005	105 (40)

Table 1: Effective diversity of number frequencies per source, against the uniform ticket-sampler null:

B=20{,}000

simulated datasets of independent uniform six-distinct-of-49 tickets at each source’s valid-ticket count, seeds 20260904 to 20260908 by sample size. The floor is the simulated 5th percentile of

N_{\mathrm{eff}}

;

p_{\mathrm{MC}}=(1+\#\{\mathrm{sim}\leq\mathrm{obs}\})/(B+1)

, one-sided, reported as

1/20{,}001=0.00005

when no simulated dataset was as concentrated. The top-number count is a descriptive diagnostic, shown with its simulated 95th percentile.

engine	valid answers	distinct tickets	modal ticket	modal share
gpt-5.6-luna	200	8	7 14 22 31 38 46	50.5%
mistral-large	188	24	7 12 19 23 31 45	26.6%
sonar (Perplexity)	200	30	7 14 23 28 35 46	68.0%
claude-sonnet-5	196	36	7 14 22 29 35 41	26.0%
gemini-3.7-flash	200	36	7 14 23 31 42 48	38.0%
grok-4.5	200	93	7 14 23 31 42 48	22.5%

Table 2: Whole-ticket behaviour per engine. Valid answers are the parsed responses of 200 attempted calls, pooled over the four prompt variants; the modal share is over valid answers.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

engine	variant	attempted	valid	distinct tickets	modal share	$N_{\mathrm{eff}}$
claude-sonnet-5	en_base	50	50	13	36.0%	10.8
claude-sonnet-5	en_lottery	50	50	17	32.0%	11.4
claude-sonnet-5	en_terse	50	50	16	26.0%	13.3
claude-sonnet-5	en_integers	50	46	13	28.3%	12.0
gemini-3.7-flash	en_base	50	50	14	62.0%	9.0
gemini-3.7-flash	en_lottery	50	50	10	64.0%	9.3

Appendix

Table 3: Per-variant results: 50 attempted calls per cell.

N_{\mathrm{eff}}

is the effective diversity of number frequencies within the cell; the modal share is over the cell’s valid answers.

engine	requested model id	returned model id	endpoint host	extra parameters
gpt-5.6-luna	gpt-5.6-luna	gpt-5.6-luna	api.openai.com	none
claude-sonnet-5	claude-sonnet-5	claude-sonnet-5	api.anthropic.com	max_tokens=128
gemini-3.7-flash	gemini-3.7-flash	gemini-3.7-flash	generativelanguage.googleapis.com	none
grok-4.5	grok-4.5	grok-4.5	api.x.ai	none
mistral-large	mistral-large-latest	mistral-large-latest	api.mistral.ai	none
sonar	sonar	sonar	api.perplexity.ai	none

Appendix

Table 4: Deployment record per engine. All calls ran on 1 September 2026, 09:05 to 09:08 UTC, one user message per call with a fresh context. No temperature, seed, or top-p parameters were sent; provider-side defaults for decoding are undisclosed by the endpoints and recorded as unknown. No search parameters were sent to any endpoint; sonar performs retrieval by default. The returned model id is the id string in each API response, uniform across all 200 calls per engine.

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Many high-stakes AI deployments proceed only if every stakeholder deems the system acceptable relative to their own minimum standard. With randomization over a finite menu of options, this becomes a feasibility question: does there exist a lottery over options that clears all stakeholders' acceptability bars? We study a query model where the algorithm proposes lotteries and receives only binary accept/reject feedback. We give deterministic and randomized algorithms that either find a unanimously acceptable lottery or certify infeasibility; adaptivity can avoid eliciting many stakeholders' constraints, and randomization further reduces the expected elicitation cost relative to full elicitation. We complement these upper bounds with worst-case lower bounds (in particular, linear dependence on the number of stakeholders and logarithmic dependence on precision are unavoidable). Finally, we develop learning-augmented algorithms that exploit natural forms of advice (e.g., likely binding stakeholders or a promising lottery), improving query complexity when predictions are accurate while preserving worst-case guarantees.

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