Abaku Combo

A combo is a combination of digits whose structure contains at least three equations of arithmetic equality.

Combos are not rare – on the contrary, there are very many of them. Pupils come across the simplest combos practically from the very beginning of Abaku-based arithmetic. All Abaku tools, techniques and teaching formats work with combos in some way.

From a pedagogical perspective, combos form stable, long-term memorable "memes" that strongly structure both the quality and the range of a learner's calculation skills.

In the phase when pupils are already playing the Abaku game, they need combos to gain higher scores, because points are counted for all Abaku equalities used within a combo in the game. Pupils therefore naturally search for and use combos, similarly to how they use an "ace" or "joker" in other games.

Sources of Combos

There are several ways to search for combos:

1. Remember a combo used by the teacher in class, by a friend in a game, or by yourself in games against bots.
2. Build your own combos using an Abaku root (an equality containing a single Abaku example).
3. Solve an Abaku logic sequence by discovering the algorithm that generates the sequence.
4. Discover algorithms for new sequences that contain combos.
5. Find new combos using a calculator.
6. Search for combos in maths–physics–chemistry tables.
7. Share and transfer combos between more and less advanced players.

1. Teaching and Play

The Abaku methodology provides teachers with a coherent set of techniques for working with combos.

Pupils are very capable of remembering an example (combo) that gives them more points or that is in some way remarkable. Most often these are three- to five-digit combos. In primary school we usually work with combinations in the range of 2–7 digits. Practical knowledge of longer combinations is limited to the truly most gifted children.

It is worth adding that for teaching and learning arithmetic, working with combinations of 7+ digits is not necessary. The number 7 (±2) is, after all, a widely cited limit on the number of independent facts (items) that can be held in short-term memory (and then, through repetition, transferred into medium- and long-term memory).

Examples

428, 8648, 9817, 1628, 22166, 97299, 98199, 61358, 71863, 55257, 32730,
86482, 74377, 37343, 13940, 24832, 38240, 15690, 66167, and many, many others.

2. Abaku Root

By analogy with language, where we work with word roots, Abaku methodology works with an Abaku root – most often a (two- or) three-digit equality. This then becomes the basis for creating combos.

Pupils first understand a three-digit "Abaku root". Very soon they also grasp a two-digit root (powers and roots) as a basis for other Abaku roots. This is why the understanding of powers and roots in Abaku is early and automatic: pupils are able to work with them from a very young age without conflicting with the traditional curriculum.

Examples of such two-digit roots:
24, 42, 28, 82, 39, 93, 11, 3437, 9729, 327, 8648, …

Examples

Abaku root 437 gives rise to these combos:
74371, 74377, 74372, 41437, 33437, 132437, 743737, 4374371, 343771, 543717, and many others.

Abaku root 636 gives rise to these combos:
6636, 66369, 63642, 963660, 563692, 216636, 663630, 1636636, 6636102, and many others.

Abaku root 177 gives rise to these combos:
17710, 17724, 81774, 78177, 77177, 76177, 177491, 177119, 166177, 871770, and many others.

Abaku root 648 gives rise to these combos:
8648, 86482, 186482, 56488, 24648, 246488, 864836, 729648, 1064858, and many others.

Abaku root 273 gives rise to these combos:
2739, 32730, 32739, 75273, 82739, 42735, 122739, 413273, 632736, 1127339, and many others.

If we swap the digits to form the root 327, we can create further combos:
9327, 39327, 393271, 393279, 32725, 32730, 32739, 932766, 141327, 632790, and many others.

3. Logic Game (Logic Sequences)

Logic sequences are combinations of digits that the teacher presents to pupils so they can discover for themselves how combos can be created. This is a playful and exploratory activity even for experienced maths teachers, because logic sequences are often built on a non-trivial principle.

Once the teacher passes on this way of thinking about combinations of digits, pupils can start inventing their own logic sequences. Teacher-built tasks then become an endless playground for number games that children enjoy regardless of their current calculation level.

Examples

Towards the combo 15510 leads the following sequence:
11110, 12210, 13310, 14410, _ _ _ _ _, 16610, 17710, 18810, 19910.
The algorithm is obvious (we systematically increase the first two digits).

Towards the combo 75273 leads the following sequence:
31228, 42240, 53251, 64262, _ _ _ _ _, 86284, 97295.
Algorithm: from a two-digit number with digit difference 2 we subtract 2 each time (in the leading part), the rest follows this pattern.

Towards the combo 81990 leads the following sequence:
9918, 18927, 27936, 36945, 45954, 54963, 63972, 72981, _ _ _ _ _.
Algorithm: we take multiples of 9 and add 9 step by step; the rest of the digits in each combo are structured accordingly.

Towards the combo 729981 leads the following sequence:
111, 824, 2739, 64416, 125525, 216636, 343749, 512864, _ _ _ _ _ _.
Algorithm: we take the 3rd power of a single-digit number and divide it by that number (i.e. we obtain its square), and embed this into Abaku form.

4. Algorithms

In building combos with Abaku logic sequences, the pupil tries to imitate their teacher. The teacher leads them to discover a certain algorithm behind an Abaku logic sequence and to check whether it contains combos. This technique works well for all age groups and is highly motivating.

Examples

Typically, pupils first discover combos created by adding, subtracting, or multiplying a two-digit number by 1.

When the two-digit number has identical digits:

• Addition:
  11112, 22123, 33134, 44145, 55156, 66167, 77178, 88189.
• Multiplication:
  11111, 22122, 33133, 44144, 55155, 66166, 77177, 88188, 99199.
• Subtraction:
  11110, 22121, 33132, 44143, 55154, 66165, 77176, 88187, 99198.

When the two-digit number has digit difference +1:

• Addition:
  21122, 32133, 43144, 54155, 65166, 76177, 87188, 98199.
• Multiplication:
  21121, 32132, 43143, 54154, 65165, 76176, 87187, 98198.
• Subtraction:
  21120, 32131, 43142, 54153, 65164, 76175, 87186, 98197.

When the two-digit number has digit difference −1:

• Multiplication:
  12112, 23123, 34134, 45145, 56156, 67167, 78178, 89189.
• Subtraction:
  12111, 23122, 34133, 45144, 56155, 67166, 78177, 89188.

5. Calculator

In Abaku methodology, the calculator is not an enemy but a useful helper. It can speed up the search for combos. We simply write down any interesting combo we find. These are mostly results of multiplying a two-digit number by a one- or two-digit number.

After some time, pupils naturally put calculators aside because they are able to calculate automatically in their head. Even then, we do not forbid calculators – they remain a good tool for independent checking. The pupil is no longer dependent on (sometimes invasive) teacher control, which reduces stress and supports independence and confidence.

Examples

• Combo 144246 comes from extending 246 (24 × 6 = 144).
• Combo 1525375 comes from 15 × 25 = 375.
• Combo 8255515 comes from 55 × 15 = 825.
• Combo 3618648 comes from 36 × 18 = 648.

6. M–P–C Tables (Maths–Physics–Chemistry Tables)

Some combos can be found directly in standard maths–physics–chemistry tables, especially those related to second and third powers of numbers.

Examples

174913
17289
341156
18324
19361

7. Sharing Combos

An individual who is gradually uncovering the "mysteries" of Abaku discovers and conquers their own combos step by step as their calculation skills grow. However, if this player uses their combos in a game against a player who has been playing Abaku for a much shorter time, the less experienced player can adopt these combos almost immediately.

Therefore:

• It is beneficial if, within a school, more advanced and less advanced players play together, across year groups.
• If this continuity is broken and a teacher builds the game culture only within one class, which then "disappears" when leaving school, it is a huge loss. Those pupils could have passed their skill on to younger ones.

Conclusion

The moment a teacher discovers that their pupils are independently developing Abaku roots into full combos, they know they are already halfway to success.

It means:

• The activity is fun for pupils.
• They understand it.
• Most importantly, it signals a fundamental shift in how pupils perceive and think about numbers.

Pupils have started to "read" numbers. This means a process has begun that is usually blocked in traditional arithmetic teaching:
the automatic reading of digit combinations and automatic searching for patterns, relations, meanings – and Abaku equalities.

At this point you can congratulate yourself: your pupils have started, willingly or not, to automatically calculate when looking at numbers, and a maths lesson will never really end for them when the bell rings.

Your Class / School Combo Bank

Build your own class or school Combo Bank – Abanka Komb. To get you started, here is a "starter capital". It is up to you whether you claim and expand this bank and how you share it.

(If this is not enough, more combo lists can be found in the methodology: Abanka komba and Pětková komba.)

Example entries:

27936
4312
2483
24832
38240
1248
1628
16824
168247
4416
441628
184260
64460
164460
8648
86482
186482
7642
76742
7428
88188
881870
87188
1881882
243862
2464
24648
246488
24648828
462470
19910
819729
97299
97295
96399
95499
93390
39336
1459451
497298199
753540
74371
74377
15690
849756
39327
393279
11920
28262
37343
46424
42850
62870
12930
9729639...
818648
497717
616366
515255
414164
853154
641648
82677749
892736
844737
544868
774986
36279339
824933
942470
552530
75273
95491
164460
497298199
82739821
24648828
1459451
243862
352844
274148
7298170
81990729
1281972
918281
684624
462468
371369
44851844
753540
71863
54963
55496
63972
46397
72981
37298
8199
81990