Concept 5 of 6
Shape Series: Two Cycles at Once
Today's big idea
Split the strip by feature, then find where both cycles land home together.
Two cycles line up again at the LCM of their lengths, not when just one returns.
How it works
👀 What's going on?
At Class 4 a shape strip has one repeating block, and finding it finishes the job. At Class 5 two things change along the strip at different speeds, and the block you are hunting for is much longer than it looks.
💡 Here's the trick
Look at the strip above. The shapes run triangle, circle, square and back again, a cycle of 3. The fill runs solid, hollow and back again, a cycle of 2. Neither cycle cares about the other. The strip as a whole only comes back to its starting look after 6 items, the smallest number both 3 and 2 divide into.
🪄 How to do it
- 1.Split the strip into its separate features, as the strip above runs a shape cycle and a fill cycle.
- 2.Find how long each cycle is on its own, then use the remainder to place the item you want.
- 3.Put the features back together, and for the whole strip use the smallest number both cycles divide into.
✍️ Try it
With shapes cycling every 3 and fills cycling every 2, what does item 10 look like? And if shapes cycled every 4 and colours every 3, after how many items would the whole strip repeat?
For item 10, count 9 steps on from the first. 9 divides by 3 exactly, so the shape is back to a triangle. 9 divided by 2 leaves 1, so the fill has moved on one and is hollow. For the second, 4 and 3 both divide into 12.
⚠️ Watch out
You might be thinking the strip repeats as soon as one feature does. It does not. The shape can be back to the start while the fill is halfway through its own cycle. The strip repeats only when both land home together.
🌟 Why this matters
Traffic lights on two junctions and two taps dripping at different rates behave this way. So does a rangoli whose shape and colour repeat on different counts. Working out when two rhythms line up again is the same question every time.
Exam tip: Write the two cycle lengths down separately before you touch the item number.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Shapes cycle every 3, fills cycle every 2. What is item 13?”
- 1
Split the two features
One cycle runs the shape, a different one runs the fill. They do not care about each other. 🔺
- 2
Count the steps, not the items
Item 13 is 12 steps on from item 1, so 12 is the number to divide by each cycle length.
- 3
Take each remainder
12 ÷ 3 leaves 0, so the shape is home at a triangle. 12 ÷ 2 leaves 0, so the fill is home at solid.
- 4
Answer ✓
a solid triangle
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Two features, two speeds. Split them before you count.
CYCLE SPLITTER · ROUND 1 OF 3
Step 1 · how do you read this strip?
Quick check
✅ VERIFICATION
Shapes cycle triangle, circle, square. Fills cycle solid, hollow. What is item 8?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Machine Rules
Undo the LAST step first, then check by running it forwards.