Depth · one concept from first principles · step 2 of 4
A white light stays on for seconds and then off for seconds, while a yellow light stays on for seconds and then off for seconds. If both lights are switched on at the same moment, for how many seconds during the first minute are both lights on at the same time?
Add the ON time and OFF time of each light to get how long before its pattern starts over. White: 9 + 6. Yellow: 7 + 3.
One ON-then-OFF round is the repeating unit; everything after just copies it, like a stamp pressed over and over.
4.OA.B.4Look For A PatternBoth cycles line up again only at the least common multiple of 15 and 10, which is 30 seconds.
After 30 seconds both lights are back to 'just turned on' together, so the next 30 seconds are an exact replay of the first 30.
6.NS.B.4Look For A PatternThe two lights' combined on-and-off pattern starts over every 30 seconds.
Each light repeats on its own fixed cycle, the white one every 15 seconds and the yellow one every 10 seconds.
The pair looks like the very start again only when both lights are back at their own starting moment together, and that first happens at 30 seconds.
A moment when both are back at the start must be a whole number of 15-second cycles and also a whole number of 10-second cycles.
From 0 to 30, white is on 0-9 and 15-24, while yellow is on 0-7, 10-17 and 20-27.
Writing out the actual ON time-blocks turns a blinking puzzle into two simple lists you can compare.
4.OA.B.4Make A Systematic ListTheir on-stretches overlap for 7, 2 and 4 seconds, giving 13 seconds in one cycle.
Solving the easier 30-second piece first means I only have to find a few short overlaps instead of scanning a whole minute.
4.OA.A.3Solve An Easier Related ProblemThe next 30 seconds replay the same pattern, so the minute holds 13 + 13 = 26 seconds.
Once one full cycle is solved, the rest of the minute is just the same answer repeated — no need to recount.
6.NS.B.4Look For A PatternWhen two things blink on their own beats, the whole pattern repeats at the least common multiple of their cycles — solve one cycle, then just repeat it.
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The same shape shows up later on the AMC, as Lattice / Grid Pattern from Small Cases, Modular Cycle Frequency Count, Units-Digit Tracking. Why the shapes repeat →