Length Rod Challenge Generator

Six challenges for a box of number rods (lengths 1 to 10, one unit per step): make a train as long as the target from the rods allowed. Each challenge asks either how many trains there are — where 2 + 3 and 3 + 2 are different, because the rods sit in a different order — or how many different sets of rods there are, where they are the same. Some challenges limit the box to one to three rods of a length. Every count on the answer sheet is exact, and the page lists every answer when there are forty or fewer.

Shortest target

2 to 24 units.

Longest target

2 to 24 units.

Question

Seed (optional)

The same seed and settings give the same sheet while the generator is unchanged.

Limit the box: some rod lengths have only 1–3 rods

Paper size

Print size

Margin

Millimetres of blank edge, 5 to 40.

Turn the paper sideways

How to print this, or save it as a PDF

There is no PDF to download, and there does not need to be. Open the print command in your browser and choose “Save as PDF” as the destination. Your browser decides whether the page address and date print across the top and bottom; if they appear, turn off “Headers and footers” in the print dialog.
Set the scale to 100%, not “fit to page”. This sheet is already the exact size of the paper you chose, so anything that resizes it makes every measurement on it wrong.
Leave the paper size in the dialog set to the same one you chose here.

Your challenges will appear here.

What this generator does

Turns the classic 'how many ways can you make ten?' rod activity into a sheet with a definite answer: the exact number of trains or sets of rods for each target, the fewest rods any answer uses, and the answers themselves when the list is short enough to check by hand.

How to use this tool

Set the shortest and longest target length (2 to 24 units).
Choose the question: trains (order matters), sets of rods (order does not), or a mix.
Tick 'Limit the box' for challenges where some lengths have only one to three rods.
Press Make challenges, check the answers, then print the two pages: challenges, then answers.

Understanding the controls

Shortest and longest target: Each of the six challenges gets a target in this range, 2 to 24 units.
Question: Trains count different orders separately; sets of rods do not; a mix gives both kinds on one sheet.
Limit the box: Gives some lengths only one to three rods, so pupils must check their box as well as the arithmetic.
Paper and seed: Paper size and density for printing; a seed rebuilds the same sheet while the generator is unchanged.

Worked examples

Target 4, rods 1–4, plenty of each: 8 trains (order matters) and 5 sets of rods (order does not): 4; 3+1; 2+2; 2+1+1; 1+1+1+1.
Target 10, rods 1–10, plenty of each: 512 trains — that is 2 to the power 9 — but only 42 different sets of rods.
Target 7 from 2s and one 3: 3 trains (2+2+3, 2+3+2, 3+2+2) but 1 set of rods; every answer uses 3 rods.

Common use cases

A number-bonds lesson with physical rods
Early combinatorics: why order doubles the count
An enrichment task where pupils must prove they found every way

How this generator works

Counts come from dynamic programming over the target length, respecting how many rods of each length are in the box. The tests list every train and every set by brute force — code that shares nothing with the counting — and compare the two for every small target. With all lengths 1 to n allowed and plenty of rods, a train of length n can be made 2^(n−1) ways, which the tests also check.

Randomness and fairness

Targets and allowed lengths are drawn with the browser's cryptographically secure generator, or from one seeded draw when a seed is typed. Seeded sheets are deterministic, not cryptographic. The counts are computed, not random.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

Rods are drawn as outlined bars with their length written in; the colour names (red for 2, orange for 10 and so on) follow a common classroom convention and are given in words only, so check them against your own set.
Printed rods are 7 mm per unit — a 70 mm scale bar is on the sheet to check the printer did not shrink it. They are a guide, not a substitute for the physical rods.
Large targets with many rods allowed have thousands of trains; the answer page gives the count and the fewest rods, not the list.