A pattern is not yet a rule
Place points around a circle and join every pair with a straight chord. Count the regions inside. With 1, 2, 3, 4 and 5 points you get 1, 2, 4, 8 and 16 regions. The doubling looks unmistakable, so the tempting assumption is that six points give 32.
The regions double each time, so six points give 32.
Six points (in general position) give 31. The doubling was a coincidence of small numbers.
The reason is countable. Each region beyond the first is created either by a chord or by a crossing of two chords. With n points there are C(n, 2) chords and C(n, 4) crossings, because any four points determine exactly one crossing. So the number of regions is 1 + C(n, 2) + C(n, 4). For six points that is 1 + 15 + 15 = 31. Once you know why, the "pattern" of powers of two loses its authority.
That is the working habit for this whole page: notice a regularity, then ask what mechanism would force it to continue.
Kaprekar's routine: a pattern you can prove by checking
In 1949 the Indian schoolteacher D. R. Kaprekar, working in Devlali near Nashik, described a simple routine. Take a four-digit number whose digits are not all equal. Arrange its digits in descending order, then ascending order, and subtract the smaller from the larger. Repeat.
Example
Start 3524 → 5432 − 2345 = 3087 → 8730 − 0378 = 8352 → 8532 − 2358 = 6174 → 7641 − 1467 = 6174
Every valid starting number reaches 6174 within at most seven steps, and 6174 then maps to itself. Here the rule is not guessed from a few cases: there are fewer than 10,000 four-digit inputs, so the claim can be verified exhaustively — and has been, many times.
Testing its limits is instructive. You must keep leading zeros: 2111 gives 2111 − 1112 = 0999, and you continue with 9990 − 0999, not 999 − 999. Drop the zero and the routine collapses. Repdigits such as 7777 give 0000 immediately, which is why they are excluded.
Try it
Numbers with fewer digits are padded with leading zeros.
Yang Hui's triangle and its Indian counterpart
The triangle of numbers in which each entry is the sum of the two above it is often called Pascal's triangle, after Blaise Pascal's 1654 treatise. Earlier versions exist. In China it is known as Yang Hui's triangle: Yang Hui printed it in 1261, attributing it to the eleventh-century mathematician Jia Xian, who used it to extract roots. In India, the commentator Halayudha in the tenth century explained a "Meru Prastara" (staircase of Mount Meru) while discussing Pingala's much older work on counting poetic metres.
Each row sums to a power of two: 1, 2, 4, 8, 16. That is a pattern with a reason. Row n counts the ways to choose items from n options, and every option is either chosen or not, giving 2n possibilities in total.
Why rows sum to powers of two — full argument
The entries of row 4 are 1, 4, 6, 4, 1. They count subsets of a four-item set with 0, 1, 2, 3 and 4 items. Every subset is counted exactly once, and there are 2 × 2 × 2 × 2 = 16 subsets, because each item is independently in or out. So 1 + 4 + 6 + 4 + 1 = 16. The same argument works for any row.
Related counting appears in Indian prosody. Counting rhythms built from short syllables (one beat) and long syllables (two beats) gives 1, 2, 3, 5, 8, 13… — a sequence discussed by Virahanka and later by Hemachandra around 1150, well before Fibonacci's 1202 Liber Abaci.
Sunzi's remainder problem
Problem · Sunzi Suanjing
There is an unknown number of things. Counted by threes, two remain. Counted by fives, three remain. Counted by sevens, two remain. How many things are there?
You must test numbers one at a time until all three conditions fit.
Combine conditions that share a remainder first; the search shrinks to a handful of candidates.
Show the worked solution
Step 1. Remainder 2 on division by 3 and by 7 means the number is 2 more than a multiple of 21. Candidates: 2, 23, 44, 65, 86, 107…
Step 2. Which leaves remainder 3 on division by 5? 2 leaves 2; 23 leaves 3. So 23 works.
Step 3 — the limit. 23 is the smallest answer, not the only one. Adding 105 (= 3 × 5 × 7) preserves every remainder, so 128, 233 and so on also fit. Sunzi's text gives 23.
Source context
The Sunzi Suanjing ("Master Sun's Mathematical Manual") is usually dated between the third and fifth centuries CE. A general method was later developed by Qin Jiushao in his 1247 Shushu Jiuzhang.
A sequence tells you what happened. A reason tells you what must happen next.
Want to see how patterns turn into choices? Continue with strategic logic, or read where these sources came from on the puzzle traditions page.