How is it built?
A construction for any antima m and sthāna n, not just m = n = 3.
What does it mean?
What the entries are, and why the flag takes this shape.
International Conference on the History of Indian Mathematics 2026
The Patākā
in the Gaṇitakaumudī
Prasad Jawalgekar · Bhumika Mittal · K. Ramasubramanian · Aalok Thakkar
Nārāyaṇa Paṇḍita and the Gaṇitakaumudī
1356
composed, CE
≈930
verses, in fourteen chapters
≈140
verses in Chapter 13 alone
The full text surfaced only in the twentieth century, when Padmākara Dvivedī found a complete manuscript in his father Sudhākara's collection and printed it in two volumes, 1936 and 1942.
Aṅkapāśa
अथ गणकानन्दकरं सङ्क्षेपादङ्कपाशकं वक्ष्ये ।
निपतन्ति यत्र मत्सरवन्तो दुष्टाः कुगणका ये ॥
atha gaṇakānandakaraṃ saṅkṣepādaṅkapāśakaṃ vakṣye | nipatanti yatra matsaravanto duṣṭāḥ kugaṇakā ye ||
I now introduce aṅkapāśa, which delights mathematicians, and where the jealous, wicked, and incompetent (kugaṇaka) suffer a fall. A pun, śleṣa: the subject of combinatorics, and a net of numbers that traps bad mathematicians.
He then lists where it applies, across six different fields:
dramatics bharata
prosody chandaḥśāstra
medicine vaidya
garland-making mālyakriyā
mathematics gaṇita
architecture śilpa
Recitation courtesy of Vāgdhenu
The Patākā antima m = 3 · sthāna n = 3
1
2
4
10
3
5
7
11
13
19
6
8
12
14
16
20
22
9
15
17
21
23
25
18
24
26
27
How do we construct a patākā for any give antima and sthāna?
The Patākā antima m = 4 · sthāna n = 3
1
2
5
17
3
6
9
18
21
33
4
7
10
13
19
22
25
34
37
49
8
11
14
20
23
26
29
35
38
41
50
53
12
15
24
27
30
36
39
42
45
51
54
57
16
28
31
40
43
46
52
55
58
61
32
44
47
56
59
62
48
60
63
64
Two Questions
The Guṇottara Sequence
आदौ रूपं विलिखेत् अन्तिमगुणितं पुरः पुनस्तद्वत् ।
स्थानाधिकं तु यावत् पङ्क्तिर्गुणकोत्तराख्येयम् ॥
ādau rūpaṃ vilikhet antimaguṇitaṃ puraḥ punastadvat | sthānādhikaṃ tu yāvat paṅktirguṇakottarākhyeyam |
Powers of m: G(m, n) = 1, m, m2, …, mn.
For m = 3: 1, 3, 9, 27, the corners that close each stage.
1
3
9
27
The Ābhyāsikī Sequence
स्थानाहतोऽन्तिमाङ्कः सैकः स्थानोनितश्च तच्छेषम् ।
आभ्यासिक्यां पङ्क्तौ प्रजायते स्थानमानमिह ॥
sthānāhato'ntimāṅkaḥ saikaḥ sthānonitaśca taccheṣam | ābhyāsikyāṃ paṅktau prajāyate sthānamānam iha |
Multiples j·mr, in increasing order: the first-row values.
1
2 3×3
6 9×3
18 27
For m = n = 3: A = 1, 2, 3, 6, 9, 18, 27, the smallest number in each column.
What is the ābhyāsikī for m = 4, n = 3?
1
2 3 4×4
8 12 16×4
32 48 64
A = 1, 2, 3, 4, 8, 12, 16, 32, 48, 64.
The Nārācikā Sequence
अन्तिममितवैश्लेषस्थानाङ्कमिताश्च ताः पृथक् स्थाप्याः ।
तासां घातः सूचीपङ्क्तिर्नाराचिका वा स्यात् ॥
antimamitavaiśleṣasthānāṅkamitāśca tāḥ pṛthak sthāpyāḥ | tāsāṃ ghātaḥ sūcīpaṅktir nārācikā vā syāt |
Coefficients of (1 + x + ⋯ + xm−1)n.
For m = n = 3: (1 + x + x²)³ = 1 + 3x + 6x² + 7x³ + 6x⁴ + 3x⁵ + x⁶, so N = 1, 3, 6, 7, 6, 3, 1.
1
2
3
6
9
18
27
Subtract One, Write in Base Three
the numbers 1 to 27
each number, less one, in three base-three digits · 1 becomes 000, 27 becomes 222
add the three digits · the digit sum runs from 0 to 6
group by digit sum, keeping each group in increasing order
the patākā assembles itself · column heights 1, 3, 6, 7, 6, 3, 1
The patākā is an enumeration of [m]n.
All mn ways to fill n places with m symbols, each set in the column of its digit sum.
The same structure surfaces in the algorithm of the verses.
The Patākā Verses
नाराचपङ्क्त्यङ्कमिताः कोष्ठानामूर्ध्वपङ्क्तयः । तिर्यग्गामी च सर्वासां स्वस्वखण्डावसानमा ॥
पङ्क्तौ तदाद्यकोष्ठे यः पल्लवोऽथाङ्कयोजनाः । तिर्यक्स्थितायामाद्यायां पङ्क्तिमाभ्यासिकीं लिखेत् ॥
तदन्तिमाङ्कः क्षेपाख्यः पुरःस्थः साध्यनामकः । क्षेपं पुरातनैरङ्कैः क्रमात् संयोजयेत् पृथक् ॥
तानधस्तिर्यग्गायां च कोष्ठपङ्क्त्यां विनिक्षिपेत् । साध्याङ्कस्य पताका स्यात् साध्यं क्षेपं प्रकल्पयेत् ॥
साध्यं पुरःस्थितं कृत्वा क्षेपं प्राग्वत् पुरातनैः । अङ्कैराद्यद्वितीयादिकोष्ठपङ्क्तिगतैर्युतम् ॥
तिर्यङ्निरङ्ककोष्ठेषु साङ्काधःस्थेषु विन्यसेत् । येनाङ्केन युतः क्षेपः साध्याङ्कादधिको यदा ॥
तदा मुक्त्वा तमङ्कं तु योजयेदितरांस्ततः ॥
Reading the Verses Again
पुरातनाङ्कः
purātanāṅka
the old number
A number already placed in the flag. Every new entry is built from one.
क्षेपः
kṣepa
the increment, j·3r
The current first-row term. Adding it raises one base-three digit by j, so the sum lands exactly j columns to the right.
साध्यः
sādhya
the bound
The next first-row term. Keep only sums below it: precisely the condition that no carry occurs.
The Patākā Algorithm
Given antima m and sthāna n: write the ābhyāsikī along the first row, then fill the grid along the diagonals.
keep track of only three numbers
kṣepa K3
sādhya S6
purātanāṅka p1, 2
first row laid · K = 3, S = 6
साध्याङ्कस्य पताका स्यात् ।
sādhyāṅkasya patākā syāt |
The patākā is [the same] as the sādhya number.
1
2
4
10
3
5
7
11
13
19
6
8
12
14
16
20
22
9
15
17
21
23
25
18
24
26
27
the interactive patākā, on this site
One Patākā Inside the Next
Each closed stage is a complete smaller patākā. Raising n by one wraps the whole flag inside a larger one.
- (3, 1)3 cells
- (3, 2)9 cells
- (3, 3)27 cells
The Algorithm, Formally
Patākā(m, n) ▷ antima m, sthāna n A ← ābhyāsikī(m, n) ▷ first-row values j·m^r write A along the first row K ← A[2]; S ← A[3] ▷ kṣepa (increment), sādhya (bound) repeat for each placed entry p, in order: if p + K < S: place p + K in the next diagonal cell ▷ column of p, shifted right by offset(K) K ← S; S ← next term of A until K = m^n return the grid offset(K) = j where K = j·m^r
Column k lists the indices of [m]n whose base-m digits sum to mn − k, in increasing order. The whole grid is built in Θ(n · mn) time, optimal for listing all mn indices.
Try it live: bhumikamittal.in/pataka-algo
Thank you
open for questions