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Rolling Along the Trail

Age 11 to 14 Short Challenge Level:

Answer: $10,15,9,20,12$

The possible scores that can be obtained are: $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$.

The third and fourth scores differ by $11$. The only pairs of numbers that do this are $(1,12)$, $(4,15)$, $(5,16)$, $(9,20)$ and $(25,36)$.

When these pairs are completed into sequences, they become:

Of these, only the fourth one uses only accessible numbers, so the sequence is $10,15,9,20,12$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.