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Altamens Investigación · Evidence review · reproducible calculation

Same Length, Fewer Chunks? Why Some Digit Lists Are Easier to Recall

Grouping and patterns let a list of digits be held as fewer units. An evidence review, plus an exact count of how often random test lists contain an easy pattern.

Altamens11 min de lectura

Resumen

A digit span score counts digits, but memory research suggests people often hold a list as a smaller number of chunks: groups recoded into one unit. Estimates of a central limit in young adults sit around three to five chunks. Lists that contain a pattern, such as 4-5-6, can be compressed, and in laboratory studies compressible lists produced longer measured spans than lists without structure.

This note reviews primary studies on chunking, grouping and trained digit memory, then adds one exact calculation. Using the digit generator in the current Altamens working memory test, 8.9% of 7-digit lists contain a counting run such as 4-5-6 and 22.1% contain a run with any equal step, such as 2-4-6. The generator already prevents repeated neighbours, which would otherwise appear in 46.9% of uniform 7-digit lists. These numbers describe list composition only: no study here measured how much these specific patterns help recall, and no Altamens user data was used.

Hallazgos clave

  • A digit span is not a direct count of memory slots. In laboratory studies, the same people recalled longer lists when the items could be grouped or recoded into fewer chunks.
  • In the current Altamens test generator, about 1 in 11 seven-digit lists contains a counting run and about 1 in 5 contains an equal-step run. Unfiltered uniform lists would repeat a neighbour in almost half of seven-digit lists.
  • Compare digit span scores only from the same test rules, and note any change in strategy. A higher span after you start grouping digits reflects the strategy on this task, not a larger memory or better memory in daily life.

1The short answer: span is counted in digits, held in chunks

Miller's 1956 paper described people remembering about seven chunks in short-term memory tasks. Cowan later reviewed a wide range of data and argued that when rehearsal and grouping are blocked, a central limit of about three to five chunks appears, averaging about four.1,2,3

A chunk is a set of items treated as one unit. If 1-9-8-4 is held as a familiar year, four digits may cost about one unit. That is why a forward digit span of seven is not proof of seven independent memory slots: part of the score can come from how the list was grouped.

2Patterns in the list raise the measured span

Mathy and Feldman manipulated how compressible stimulus sequences were by adding sequential patterns of different lengths. Their participants' measured digit span consistently depended on the length of the list after compression. They estimated the true limit at about 3 or 4 distinct chunks, equivalent to about 7 uncompressed items of typical compressibility.4

Chekaf, Cowan and Mathy built lists with possible associations between items. Untrained participants recalled about three items when a list had no structure, and recall rose linearly as structure was added.5

Neither study used exactly the patterns counted in section 5 of this note, so their effect sizes cannot be transferred to the Altamens test. They do show the direction: structure inside a list can lengthen the span someone appears to have.

3How chunks help, and where the benefit stops

Thalmann, Souza and Oberauer found across four experiments that chunking helped recall of the chunked material and also of other, unchunked material held at the same time, which supports the idea that chunks reduce load. Chunks early in a list helped later material; chunks at the end did not. The benefit was not independent of chunk size in every condition, so capacity is not simply a fixed number of chunks of any size.6

Norris, Kalm and Hall found that small two-word chunks behaved as if long-term memory helped rebuild a degraded trace, while three-word chunks behaved more like data compression. Norris and Kalm's review argues that chunking can use short-term capacity more efficiently, but that capacity is not set by the number of chunks alone.7,8

Rhythm matters too. Studies of immediate serial recall report grouping effects when items are spaced in time, for mid-length lists. Reading a phone number out in groups is an everyday example of the same kind of spacing. Hartley, Hurlstone and Hitch found these temporal grouping effects did not depend on whether the grouping pattern was predictable.9,10

4Trained digit memory: a large gain on the practised task

The best-known demonstration is a single participant who practised digit span in a laboratory. After more than 230 hours, his span rose from 7 to 79 digits, and his performance on other memory tests with digits equalled that of memory experts with lifelong training. The authors attributed this to an appropriate mnemonic system.11

Gobet and colleagues distinguish deliberate, strategic chunking from automatic chunking linked to perception and learning. The trained case is deliberate recoding of one kind of material, measured with tests of that material. It shows how far a practised strategy can move a span score; it does not show a general change in memory, and this note makes no claim of that kind.12

5How often a random test list contains an easy pattern

We inspected the digit generator of the Altamens working memory test (makeDigits in app/tests/working-memory/page.tsx, commit 6dd71de). Each list starts at 3 digits and grows by one after every correct answer. The first digit is drawn uniformly from 0 to 9, and each later digit is drawn uniformly from the 9 digits that differ from the previous one. A code comment gives the reason: immediate repeats are disproportionately easy to chunk. The generator does not filter other patterns.

We counted three simple, adjacent patterns: a repeat (5-5), a counting run of three digits stepping by +1 or -1 (4-5-6, 8-7-6) and an equal-step run of three digits with any shared non-zero step (2-4-6, 9-6-3, which includes counting runs). Probabilities were computed exactly by dynamic programming, then checked against brute-force enumeration up to 6 digits and a seeded Monte Carlo run of 200,000 lists per length.

Figura 1Share of lists containing at least one pattern, by list length
Uniform: repeated neighbourAltamens generator: equal-step runAltamens generator: counting run
-4.722.048.675.335781012Uniform: repeated neighbour · 3 digits in the list: 19%Uniform: repeated neighbour · 5 digits in the list: 34.4%Uniform: repeated neighbour · 7 digits in the list: 46.9%Uniform: repeated neighbour · 9 digits in the list: 57%Uniform: repeated neighbour · 11 digits in the list: 65.1%Uniform: repeated neighbour · 12 digits in the list: 68.6%Altamens generator: equal-step run · 3 digits in the list: 4.9%Altamens generator: equal-step run · 5 digits in the list: 13.9%Altamens generator: equal-step run · 7 digits in the list: 22.1%Altamens generator: equal-step run · 9 digits in the list: 29.5%Altamens generator: equal-step run · 11 digits in the list: 36.1%Altamens generator: equal-step run · 12 digits in the list: 39.2%Altamens generator: counting run · 3 digits in the list: 2%Altamens generator: counting run · 5 digits in the list: 5.5%Altamens generator: counting run · 7 digits in the list: 8.9%Altamens generator: counting run · 9 digits in the list: 12.2%Altamens generator: counting run · 11 digits in the list: 15.4%Altamens generator: counting run · 12 digits in the list: 17%digits in the list
lists with the pattern (%)

Every length from 3 to 12 is in results.csv. Lines connect the plotted lengths for readability.

Exact probabilities from calculate.py. 'Altamens generator' forbids repeated neighbours; 'uniform' draws every digit independently. These are properties of the lists, not measured effects on recall.

With the current generator, 8.9% of 7-digit lists contain a counting run and 22.1% contain an equal-step run. At 9 digits the shares are 12.2% and 29.5%; at 12 digits, 17.0% and 39.2%. Without the repeat filter, 46.9% of uniform 7-digit lists would contain a repeated neighbour, and 58.6% would contain a repeat or an equal-step run. Removing repeats slightly raises the share of counting runs, from 7.3% to 8.9% at 7 digits, because the probability moves to the other digits.

Download the results table (CSV), the standard-library script and the methods and claim sheet. The files contain no participant or customer data.

6How to read and compare a digit span score

  1. Compare scores from the same test only. Start length, digit timing, the number of misses allowed and whether patterns are filtered all change what a span means.
  2. Expect some run-to-run variation from the lists themselves. In a randomised test, one attempt may happen to include a 4-5-6 or 2-4-6 run and the next may not.
  3. Write down your strategy. Grouping in threes, saying the digits aloud or recoding them as dates are legitimate strategies, but a score with a new strategy is not comparable with an old score without one.
  4. Do not convert a span in digits into a capacity in chunks or a percentile. This note gives no norms, and the Altamens test reports a personal task score, not a population ranking.
  5. If you design a test, decide which patterns to allow and say so. Filtering repeats, as the Altamens test does, removes the most common easy pattern; filtering runs as well would remove another source of difference between lists, at the cost of lists that are less random.

Try the working memory test with the rules above in mind. For how much a digit span moves between two consecutive blocks under different scoring rules, see How Stable Is Your Digit Span? For grouping habits you can use while playing, see the memory games guide.

7Limitations, provenance and review status

The pattern definitions are simple rules chosen for transparency. They are not the compression measures used in the cited studies, and people may chunk other structures (dates, familiar numbers, rhythm) that the count ignores. The calculation measures how often patterns occur, not how much they help. The literature review is targeted rather than systematic, several summaries rely on study abstracts, and the laboratory samples were not drawn from Altamens players.

No Altamens user records, accounts, analytics, devices or game telemetry were accessed. The only Altamens input is the public source code of the test generator. Literature search, calculation, drafting, source checks and publication preparation used an AI assistant. The script and the displayed figures received automated consistency checks, but the paper has no independent human methods review and is not journal peer reviewed.

Referencias

  1. 1.Miller GA The magical number seven plus or minus two: some limits on our capacity for processing information. Psychological Review 63(2), 81-97, 1956. Classic summary. DOI: 10.1037/h0043158.. https://pubmed.ncbi.nlm.nih.gov/13310704/
  2. 2.Cowan N The magical number 4 in short-term memory: a reconsideration of mental storage capacity. Behavioral and Brain Sciences 24(1), 87-114, 2001. Target article reviewing capacity limits of three to five chunks. DOI: 10.1017/S0140525X01003922.. https://pubmed.ncbi.nlm.nih.gov/11515286/
  3. 3.Cowan N The magical mystery four: how is working memory capacity limited, and why?. Current Directions in Psychological Science 19(1), 51-57, 2010. Open-access overview. DOI: 10.1177/0963721409359277.. https://pmc.ncbi.nlm.nih.gov/articles/PMC2864034/
  4. 4.Mathy F, Feldman J What's magic about magic numbers? Chunking and data compression in short-term memory. Cognition 122(3), 346-362, 2012. Experiments manipulating the compressibility of digit sequences. DOI: 10.1016/j.cognition.2011.11.003.. https://pubmed.ncbi.nlm.nih.gov/22176752/
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  6. 6.Thalmann M, Souza AS, Oberauer K How does chunking help working memory?. Journal of Experimental Psychology: Learning, Memory, and Cognition 45(1), 37-55, 2019. Four experiments. DOI: 10.1037/xlm0000578.. https://pubmed.ncbi.nlm.nih.gov/29698045/
  7. 7.Norris D, Kalm K, Hall J Chunking and redintegration in verbal short-term memory. Journal of Experimental Psychology: Learning, Memory, and Cognition 46(5), 872-893, 2020. Six experiments. DOI: 10.1037/xlm0000762.. https://pubmed.ncbi.nlm.nih.gov/31566390/
  8. 8.Norris D, Kalm K Chunking and data compression in verbal short-term memory. Cognition 208, 104534, 2021. Review. DOI: 10.1016/j.cognition.2020.104534.. https://pubmed.ncbi.nlm.nih.gov/33360054/
  9. 9.Spurgeon J, Ward G, Matthews WJ, Farrell S Can the effects of temporal grouping explain the similarities and differences between free recall and serial recall?. Memory & Cognition 43(3), 469-488, 2015. Lists of 1 to 12 words. DOI: 10.3758/s13421-014-0471-5.. https://pubmed.ncbi.nlm.nih.gov/25331276/
  10. 10.Hartley T, Hurlstone MJ, Hitch GJ Effects of rhythm on memory for spoken sequences: a model and tests of its stimulus-driven mechanism. Cognitive Psychology 87, 135-178, 2016. DOI: 10.1016/j.cogpsych.2016.05.001.. https://pubmed.ncbi.nlm.nih.gov/27261540/
  11. 11.Ericsson KA, Chase WG, Faloon S Acquisition of a memory skill. Science 208(4448), 1181-1182, 1980. Single-participant case study.. https://doi.org/10.1126/science.7375930
  12. 12.Gobet F, Lane PC, Croker S, Cheng PC, Jones G, Oliver I, Pine JM Chunking mechanisms in human learning. Trends in Cognitive Sciences 5(6), 236-243, 2001. Review. DOI: 10.1016/S1364-6613(00)01662-4.. https://pubmed.ncbi.nlm.nih.gov/11390294/