Gaps & Intervals
Count the spaces, not the posts — always off by one.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: When objects sit in a line with one at each end, the spaces between them are always one less than the objects.
★ · If N objects sit in a line with one at each end, then there are N − 1 spaces between them — always one less.
Hold up one hand with your fingers spread out. There are 5 fingers in a row, with one finger at each end. How many gaps sit between neighbouring fingers?
L0 · If N objects are equally spaced from end to end, then gap = total ÷ (N − 1) — divide by spaces, not by objects.
Student volunteers set up 8 rest mats in a row inside a school gym. There is a mat at each end. The spaces between mats are all equal. From the first mat to the last mat there are 28 floor tiles. How many tiles sit between two neighbouring mats?
L1 · If equally-spaced objects are numbered 1..N, then the distance from object i to object j is |j − i| × gap — count the gaps you cross, not the stops.
After a storm, a school becomes a community relief hub. Volunteers set up 10 water stations equally spaced along a 54 m walking route. One station sits at the entrance and one sits at the gym. What is the distance from the 3rd station to the 6th station?
L2 · If departures (or strikes / events) are at equal time intervals with both endpoints known, then (last − first) ÷ (N − 1) = interval. Times are points on a line.
A shuttle bus takes families to a clinic at equal time intervals. The first shuttle departs at 8:12 and the last shuttle departs at 9:48. There are 9 total trips. How many minutes apart are neighbouring shuttles?
L3 · If a regular pattern would place N₀ objects but some positions are blocked (end clearances, excluded zones), then count = N₀ − (blocked); subtract, never recount.
Volunteers light a 120 m outdoor path with lanterns every 6 m, with a lantern position at each end. There are 2 restrictions: keep 12 m clear at each end for wheelchairs (no lanterns within 12 m of either end), and no lanterns in the medical zone from 54 m to 72 m. How many lanterns can be placed?
L4 · If two plans (different spacing, different count) both fit the same fixed corridor, then (N₁ − 1) × g₁ = (N₂ − 1) × g₂ — equate spaces × gap.
Volunteers place chairs along a school corridor for tired families. Two plans cover the SAME corridor. In both plans, the first chair sits 2 m from one end and the last chair sits 2 m from the other end. Plan A uses 10 chairs spaced x m apart. Plan B uses 7 chairs spaced (x + 3) m apart. Find the length of the corridor.
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Posts and Spaces
Peek the trick — Posts and Spaces
Whenever objects sit in a line with one at each endpoint, the gaps between neighbours are always one fewer than the objects themselves. This single off-by-one move is the foundation for every level that follows.
L0 · Count the Spaces
Peek the trick — Count the Spaces
When you know the total length and the count of equally spaced objects with one at each end, divide the length by the number of spaces — which is one less than the count — not by the number of objects.
L1 · Position to Position
Peek the trick — Position to Position
Between two numbered stops in an evenly spaced row, the distance is the difference of the position numbers times one gap. Count the gaps crossed, never the stops.
L2 · Time-Posts
Peek the trick — Time-Posts
Equal-interval events are points on a number line of time, so the spacing equals the total time span divided by one fewer than the number of events. The off-by-one move applies to clocks just as it does to mats.
L3 · Pattern Minus Forbidden
Peek the trick — Pattern Minus Forbidden
Start by counting the full unrestricted pattern as if nothing were blocked, then subtract the positions removed by end clearances and forbidden zones. It is faster and safer than recounting the surviving positions one at a time.
L4 · Same Span, Two Plans
Peek the trick — Same Span, Two Plans
When two layouts must cover the same total span, write each layout's length as spaces times gap and set the two expressions equal. One span described two ways gives you the equation that unlocks the unknown.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
