In the two rows below, colours repeat as shown. When will the blue stars meet next?

Question

Homi · Accepted Answer

We need to find when blue stars meet again.

Step 1 — Analyze the first row pattern

Let us look at the first row of stars. The colors repeat in a sequence. The sequence is Yellow, Green, Orange, Blue, Pink, Grey. This sequence has 6 different colors. The blue star is the 4th star in this sequence. The blue star repeats every 6 positions. We can write these positions as: $4, 10, 16, 22, 28, \dots$

Diagram 1

Step 2 — Analyze the second row pattern

Now let us look at the second row of stars. The colors also repeat in a sequence. The sequence is Green, Orange, Yellow, Blue. This sequence has 4 different colors. The blue star is the 4th star in this sequence. The blue star repeats every 4 positions. We can write these positions as: $4, 8, 12, 16, 20, \dots$

Diagram 2

Step 3 — Find the common meeting positions

Both rows have a blue star at position 4. This is their first meeting point. We need to find when they meet next. First row blue stars repeat every 6 positions. Second row blue stars repeat every 4 positions. We find the Least Common Multiple (LCM) of these cycles. We need to find the LCM of 6 and 4.

Let us list multiples of 6: $6, 12, 18, 24, \dots$ Let us list multiples of 4: $4, 8, 12, 16, 20, \dots$ The smallest common multiple is 12. So, the LCM of 6 and 4 is: $\text{LCM}(6, 4)$ $= 12$

$\boxed{12}$

Step 4 — Calculate the next meeting position

The LCM of 6 and 4 is 12. This means the blue stars will align every 12 positions. They first met at position 4. We add the LCM to the first meeting position. Next meeting position = First meeting position + LCM. $4 + 12$ $= 16$

$\boxed{16}$ The blue stars will meet next at the 16th position.

Answer

The blue stars will meet next at the 16th position.

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