Counting Yarn Between the Loops

The front of punch needle embroidery keeps a secret: all those plush loops began as small, deliberate movements on the other side. Stretch the backing, thread the needle, settle into consistent depth and spacing. Then discover that the patch of colour you want to finish sits somewhere in the middle of your yarn budget.

Suppose four rows contain 12, 15, 10 and 13 stitches. For a deliberately simple model, give each stitch two 5 mm loop legs plus 3 mm of travel: 13 mm altogether. That leaves out tails, fabric thickness and tension effects; it is a sketch for comparing rows, not a shopping estimate.

A prefix sum records how much yarn we have accounted for at every row boundary:

Row boundary:       0    1    2    3    4
Yarn so far (mm):   0  156  351  481  650

Rows two through three need 481 − 156 = 325 mm. The leading zero gives the first row a boundary too, so every interval follows the same rule.

Prolog carries the running total through a recursive relation. Save this as loops.pl and run swipl loops.pl:

:- initialization(main, main).
prefix([], _, []).
prefix([N|Ns], Before, [After|Rest]) :-
    After is Before + N * 13,
    prefix(Ns, After, Rest).
main :-
    prefix([12,15,10,13], 0, Totals), P = [0|Totals],
    nth0(1, P, Start), nth0(3, P, End),
    Length is End - Start,
    format('~w~nRows 2-3: ~d mm~n', [P, Length]).

Go builds the same boundary totals in a slice. Save it as loops.go and run go run loops.go:

package main

import "fmt"

func main() {
    stitches := []int{12, 15, 10, 13}
    prefix := make([]int, len(stitches)+1)
    for i, count := range stitches {
        prefix[i+1] = prefix[i] + count*13
    }
    length := prefix[3] - prefix[1]
    fmt.Println(prefix)
    fmt.Printf("Rows 2-3: %d mm\n", length)
}

Building the totals takes one pass. Afterward, Go answers any contiguous row-budget question with two indexed reads and a subtraction. Prolog’s list indexing still walks the list, so the arithmetic shortcut does not automatically deliver constant-time access there.

There is another catch hiding in the backing: change one row’s stitch count and every later total needs updating. Prefix sums favour settled designs and repeated questions. Once the spacing is decided, the accounting becomes pleasantly quiet.