6^6 = (6^3)^2 = 216^2 = 46656

February 22, 2026

6^6 = (6^3)^2 = 216^2 = 46656

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5Question: A climatologist analyzing temperature anomalies over 7 consecutive years observes that each year’s anomaly is an integer between $-3^\circ C$ and $3^\circ C$, inclusive. How many distinct sequences of anomalies are possible if no two consecutive years can have the same anomaly?
Solution: Each year’s temperature anomaly has $7$ possible integer values: $-3, -2, -1, 0, 1, 2, 3$. For the first year, there are $7$ choices since no restriction applies. For each subsequent year, the anomaly must differ from the previous year’s, so there are $6$ choices. Since there are $7$ years and only the first choice is unrestricted, the total number of valid sequences is:
\times 6^{6}
\times 46656 = 326592
Thus, the number of distinct sequences is $\boxed{326592}$.
Question: A home-schooled student simulates genetic mutations using 3 fair 6-sided dice, each representing a gene locus. What is the probability that exactly two of the three dice show a prime number?

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