Predicts the next number in a given sequence of numbers.
This project was inspired by the NPAT tests that applicants need to take for certain universities or companies. The test presents number sequences where you have to figure out the next number. Machine Learning will not easily work here because some sequences may contain only 3 numbers and you have to guess the simplest next number. And some patterns are quite hard, which will require a huge amount of training data to train a model, whereas you can simply write down the formula and be done with it.
Here are some guidelines that I am following in order to simplify ambiguous cases:
- RULE 1: A single pattern should find at least 3 values (ref RULE 4) Reduce the constraint to 2 values if another pattern is found in the series (eg splitting the series into 2, one pattern finds 3 values, another finds 2)
- RULE 2: How do we determine complexity of a pattern: common standard series, +1, -1, *2, /2, ^2, recursive level, or something else A pattern (eg. diffs) with all the same numbers (3,3,3,...) is simpler than a pattern such as (7,9,13,...), but the 2nd pattern may be discovered before the 1st. 2, 5, 3, 6, 5, 8, 6, x So, it means we need to check multiple patterns before settling on a single one.
- RULE 3: What shoud the depth of recursion be? Keep it 2 for now.
- RULE 4: Should the rules be same for all levels of recursion? Heuristially, it is likely to find a simple standard series in a level 1 or 2 in a recursive call.
- RULE 5: Assume series starts at the 1st number for now. We can generalize this later.
- RULE 6: Should we consider a subset/subsequence of a standard series? For now, let's consider both.
- RULE 7: Are the basic opearations (+,-,*,/) in one direction enough to find a pattern. No! Because we are also matching standard series, so, we need to match 1,1/2,1/3... or 1/2,1/3,1/5,1/7... for eg, which becomes tricky unless we simply reverse the division.
Types of patterns implemented:
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Standard series: (Natural): 1 2 3 4 5 6 (Primes): 2 3 5 7 11 etc...
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Standard series interleaved: 1, 2, 9, 24, 120, 36 In the above series factorials and squares are interleaved: 2 24 120, 1 9 36
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Diffs: 1 4 7 10 13 16 The above series has a constant diff of 3
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Diffs of Diffs: 2, 3, 6, 13, 28; diff1: 1, 3, 7, 15 diff2: 2, 4, 8, (16) => exponentials
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Interleaving: 1, 9, 0, 2, 7, 1, 3, 5, 2, 4, 3 The following three series are inteleaved to form the above series: series1: 1 2 3 4 ... series2: 9 7 5 3 ... series3: 0 1 2 ...
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Padded: 7, 26, 63, 124 The above series is composed of padded cubes: 2^3-1, 3^3-1, 4^3-1, 5^3-1
2, 3, 10, 15, 26 The above series is composed of alternating padded squares: 1^1+1, 2^2-1, 3^2+1, 4^2-1, 5^2+1
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2-group: 2, 5, 3, 6, 5, 8, 6 (2, 5), (3, 6), (5, 8), (6, x) => in each group a-b = 3
3, 4.5, 2, 3, 7.5, 11.25, 6.2 (3, 4.5), (2, 3), (7.5, 11.25), (6.2, x) => in each group a*1.5 = b
2, 4, 12, 14, 42, 44 primay grouping: (2, 4), (12, 14), (42, 44) => in each group a-b = 2 secondary grouping: 2, (4, 12), (14, 42), (44, x) => in each group a*3 = b
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2-group-interleaving
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3-group: 2, 1, 3, 3, 2, 5, 9, 0, 9, 1, 9 (2, 1, 3), (3, 2, 5), (9, 0, 9), (1, 9, x) => in each group a+b=c
1, 8, 12, 2, 2, 3, 4, 2, 0.75, 4, 3 (1, 8, 12), (2, 2, 3), (4, 2, 0.75), (4, 3, x) => in each group a^b*c=12
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3-group-interleaving
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4-group: 56, 89, 33, 21, 57, 88, 35, 19, 60, 85, 30 (56, 89, 33, 21), (57, 88, 35, 19), (60, 85, 30, x) => in each group a+b+c+d=199
5, 1, 2, 8, 8, 1, 0, 9, 9, 2, 3 (5, 1, 2, 8), (8, 1, 0, 9), (9, 2, 3, x) => in each group a+b+c=d
35, 5, 7, 12, 14, 2, 7, 9, 42, 6, 7 (35, 5, 7, 12), (14, 2, 7, 9), (42, 6, 7, x) => in each group a=b*c, b+c=d
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4-group-interleaving
TODO: Power series TODO: Use TPL