Design: n identical bars of height a, side by side. There's nothing to rearrange — the bars already are an n×a grid of unit boxes the moment they're placed next to each other. This is the simplest possible case and it's shown first so the "boxes = area = sum" idea is established before anything gets rearranged.
What it means: multiplication itself is just repeated addition laid out as a rectangle. Every other pattern in this tool is really asking "what shape does THIS series' bars naturally rearrange into?" — and the constant case is the trivial answer: they're already there.
Design: the staircase of ascending bars is duplicated, flipped, and stacked on top of itself. Term i (blue, height aᵢ) always sits under term (n+1−i) (pink, height an+1−i), and every column’s combined height is the same constant, a₁+aₙ — that's the whole trick young Gauss is said to have found. A dashed bounding box is drawn around the result so you can see it really is a plain n×(a₁+aₙ) rectangle, no gaps.
What it means: 2Sₙ = n(a₁+aₙ), so Sₙ = n(a₁+aₙ)/2. This is the single most important rearrangement in the whole set — Natural numbers and Even numbers below are just this same picture with different a and d plugged in, which is deliberate: seeing the identical shape reappear under a different name is the point, not a coincidence to gloss over.
The AP staircase with a=1, d=1. Kept as its own button (rather than folded silently into AP) because 1+2+…+n is the identity every student meets first, and it deserves to be recognised on sight as "just an AP" once you've seen the general case.
The same AP staircase again, a=2, d=2 — exactly double the Natural-number bars. Placing it right after Natural, using the identical Gauss-pairing geometry, makes the "these are all one shape" point land without saying it outright.
Design: plain increasing bars, one colour per term so the exponential jump in height between consecutive bars is visible at a glance — and deliberately no attempted rearrangement into a rectangle, because there isn't one. Bar heights are capped and rescaled (never the raw values, which blow up fast: a=5, r=3, n=8 already exceeds ten thousand) so the shape stays legible while the printed value stays exact.
What it means: since there's no tiling proof, the tool shows the actual textbook technique — multiply the whole sum by r, subtract the original sum, and every middle term cancels (telescopes), leaving just rSₙ−Sₙ = arⁿ−a. This is included specifically so the tool doesn't pretend every series has a picture-proof; GP is the honest counter-example, proved algebraically instead.
Design: each odd number 2k−1 is built as an L-shaped band (a "gnomon") — one flat tile-strip along the top, one down the side — sized so it exactly wraps a (k−1)×(k−1) square into a k×k square. Stack all n gnomons in order and, tile by tile, they interlock into one solid n×n square with no gaps and no overlaps.
What it means: this is a genuine visual proof, not an illustration of one — the boxes really do compose the square, so 1+3+5+…+(2n−1)=n² is watched happening rather than asserted. It's the direct geometric ancestor of the "square number" idea, which is why Squares (below) is placed right after it.
Design: each term is shown as a literal k×k grid of unit tiles — a genuine square, unlike the Odd-number gnomons which were parts of a square. All n squares are laid out side by side on a shared baseline so you can compare their areas directly. Deliberately, they are not forced into one big rectangle, because 1²+2²+…+n² doesn't tile flat the way the AP or Odd-number series do.
What it means: the closed form n(n+1)(2n+1)/6 is real, but it isn't "obvious from a picture" the way Sₙ=n² is for odd numbers — it's normally reached by stacking three offset staircases of squares into a box, or by induction. Showing the squares honestly as separate, un-tiled shapes is the point: it signals to the student that this one needs the algebraic argument, not just a rearrangement.
Design: the same L-shaped gnomon idea as Odd numbers, but scaled by triangular numbers instead of unit steps — band k grows a square of side Tₖ−₁ (where Tₖ = 1+2+…+k) up to side Tₖ, and that band's area works out to exactly k³. Reusing the identical two-piece L-band construction from the Odd-number proof, just re-scaled, is intentional: it shows the same rearrangement trick generalises rather than being a one-off coincidence.
What it means: stack all n bands and the whole figure is one solid square of side (1+2+…+n) — so 1³+2³+…+n³ = (1+2+…+n)² = Tₙ², the famous "sum of cubes is the square of the sum" identity, made literally visible as one square built from n interlocking bands.
Design: the blue staircase (row r has r+1 tiles) is already the triangular number Tₙ by itself. A second, identical staircase rotated 180° (pink) is placed in the gap it leaves, and together the two pieces tile a solid n×(n+1) rectangle exactly — the same doubling trick as the AP proof above, but applied to a 2-D staircase instead of a 1-D bar chart.
What it means: since the two staircases are congruent, each is exactly half the rectangle, giving Tₙ = n(n+1)/2 — and this Tₙ is exactly the quantity that reappears as the side length of the square in the Cubes proof above, tying the two shapes together.
Design: each new square's side length is built as the sum of the two previous squares' sides, and placed flush against them, spiralling outward (right, up, left, down, repeating). This is the one pattern in the set defined by a recurrence rather than a formula in n, so the design goal shifts: instead of proving a summation identity, the geometry exists to make the recurrence aₙ=aₙ−₁+aₙ−₂ visible as a physical fact — you can watch each square being exactly large enough to tile flush with no gap or overlap against the two before it.
What it means: there's no simple closed form for aₙ itself, and none is claimed. What the tool does prove, and states as a running total, is that the sum of the first n Fibonacci numbers always equals F(n+2)−1 — included so the mode doesn't leave Fibonacci as "just a pretty spiral" but ties it back to the same Sₙ-tracking every other pattern does.
The deliberate design choice across all ten is not to force every series into the same "boxes rearrange into a rectangle" mould. Six of them (Constant, AP, Natural, Even, Odd, Cubes, Triangular) genuinely do have a clean tiling proof, and the geometry is the argument. Three (GP, Squares, Fibonacci) don't, and the tool says so in the note text and shows the real technique used in practice — telescoping, staircase-stacking/induction, and the recurrence itself. A tool that always finds a rectangle, even for series that don't have one, teaches a false lesson about when visual proofs work.