# VectorPaint five-row Pascal triangles: check adjacent sums and row totals

Language: en
Canonical: [VectorPaint five-row Pascal triangles: check adjacent sums and row totals](<https://allluckly.cn/en/guides/vectorpaint-five-row-pascal-adjacent-sums/>)
中文：[中文 HTML](<https://allluckly.cn/guides/vectorpaint-five-row-pascal-adjacent-sums/>)
English: [English HTML](<https://allluckly.cn/en/guides/vectorpaint-five-row-pascal-adjacent-sums/>)
Published: 2026-11-09T00:00:00+08:00
Modified: 2026-11-09T09:37:10+08:00
Author: Bison

Arrange fifteen entries from the top row and check boundary ones, adjacent sums and five row totals without assuming automatic formula generation.

English translation of the [Chinese original](<https://allluckly.cn/guides/vectorpaint-five-row-pascal-adjacent-sums/>), translated on 2026-11-09. Original publication dates, attribution, code and version notes are retained.

Triangular arrangement alone does not establish Pascal-triangle relations. Begin with top-row1 and draw five rows, checking boundary ones and the two parents of every interior entry. This manually verified rule diagram assumes no formula engine, automatic table filling or proof tool.

## Count the top line among the five rows

Rows may be indexed0–4 or1–5, but declare the convention. Display first through fifth, including the top1. Rows are1;1,1;1,2,1;1,3,3,1;1,4,6,4,1. Preserve this external list and count1+2+3+4+5=15 positions. Keep row labels outside the sums. A source’s “fourth row” needs its indexing convention checked before declaring an error. Historical naming/discovery order is outside scope.

## Use fixed row spacing and half-step offsets

Put the top at(600,200), row spacing120 and within-row spacing120. Row2 usesx540,660; row3 x480,600,720, continuing downward around centre600. Do not use random text-bound centring that breaks parent relationships. Independent text objects can be checked against temporary position marks before font choice. Use one positional rule, without perspective-scaling upper digits smaller and lower digits larger.

## Fix both boundaries to one

Fill each row’s left/right boundary with1, counting the lone top only once rather than overlaying two ones. This exercise defines boundary ones directly without assuming software adds unseen outside zeroes. Inspect both edges; a mistaken2 at row3’s left is not excused by an interior2 elsewhere. Temporary pale boundary boxes may be removed after checking; final readability should not require colour. Filling boundaries before interiors reduces misplaced row copying.

## Trace two parents for every interior entry

Row3’s interior2 comes from1+1; row4’s threes from1+2 and2+1; row5’s4,6,4 from1+3,3+3,3+1. Identify the two immediately adjacent positions above before entering each result, not arbitrary same-row numbers. Show two fine parent leaders for the central6 instead of a dense web. Leaders are explanatory objects, not live text dependencies, so changing parents does not recalculate descendants.

## Use row totals as a second check

Row sums are1,2,4,8,16, doubling each time. Keep totals in a separate check table rather than appending another triangle entry. A correct sum does not replace entry checks because opposite errors may cancel. Verify parents first, then use totals for omissions/duplicates. Distinguish five rows, fifteen positions and final-row sum16; the fifth row has five entries, not sixteen. Label those quantities explicitly.

## Test a copy with central six changed to seven

Save a wrong copy changing only row5’s central6 to7. It fails3+3 and changes row total16 to17. Preserve the correct version; do not compensate by changing a neighbouring4 to3. An optional sixth row must be explicitly added and separated as1,5,10,10,5,1, summing32. It is no longer the five-row diagram, and a two-digit10 is one entry rather than two positions.

## Inspect digits and adjacency in output

Save and open a PNG, reading five rows/fifteen entries, particularly3,4,6 under font/overlap changes. With Arabic digits distinguish1 from lowercase l; any10 remains one text entry. Hide unwanted position marks/check boxes actively. For crowded long entries adjust spacing/font consistently rather than compressing one10 alone. A thumbnail should retain centring/offset relationships, while detailed sums can sit beside it rather than inside every position.

## Check row numbering, boundaries, parents and totals

For an extra row check whether top1 was excluded; for a wrong interior value inspect its two parents; for a slanted layout inspect half-step offsets instead of hiding it under a thick triangular frame. Correct totals with wrong parents still require entry-level errors to remain visible. Retain rows, fifteen-position layout, central explanation,counterexample and output, identifying an optional sixth row separately. This verifies manual rule expression, not a binomial-theorem proof, infinite sequence generation or universal computation validation.

## Related tutorials and sources

[VectorPaint order-four Latin squares: check one to four in every row and column](<https://allluckly.cn/en/guides/vectorpaint-four-order-latin-square/>), [VectorPaint text layout: make a clear event card](<https://allluckly.cn/en/guides/vectorpaint-text-layout/>).

Checked on November 9, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Capability scope follows the [official App Store listing](<https://apps.apple.com/cn/app/id1587272880>) and [Apple public version lookup](<https://itunes.apple.com/lookup?id=1587272880&country=cn>). This is an original exercise to perform independently; use controls in your installed version. Manual construction and external records do not imply new automatic analysis features. No step-by-step physical-device, private-file or all-output-environment testing is claimed.