MathDB

3

Part of 2024 JHMT HS

Problems(2)

Painted Cube Probability

Source:

9/7/2024
Amelia has 2727 unit cubes. She selects one and paints one of its faces. She then randomly glues all 2727 cubes together to form a 3×3×33 \times 3 \times 3 cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
probability2024
Circling Positive Integers

Source:

9/8/2024
Let N2N_2 be the answer to problem 2.
On a number line, Tanya circles the first ℓ\ell positive integers. Then, starting with the greatest number in the most recent circle, she circles the next ℓ\ell positive integers, so that the two circles have exactly one number in common; she repeats this until N2N_2 is in a circle. Compute the sum of all possible values of ℓ\ell for which N2N_2 is the greatest number in a circle.
number theory2024