Time Is Power!

While taking a coffee break, my secretary, Sue, happened to glance at her digital watch. It showed the following time:

1 : 4 4

“That’s rather curious,” she thought. “If I remove the two dots, I’ll have the number 144, which is the square of 12. I wonder how likely that sort of thing happens during my workday?”

Assuming that Sue works from 7 a.m. until 7 p.m., what is the probability that her watch will show a square number? Express your answer to the nearest hundredth of a percent.

EXTRA: what is the probability that a cube, 4th power, 5th power, etc. will occur in addition to squares?

SUPER-EXTRA: what is the probability for a power of any kind appearing on a watch set to a 24-hour day? (This means, times can range from 0:00 to 23:59. And “first” powers are not allowed throughout this problem.)

HYPER-EXTRA: if we now use the “seconds” digits that appear on many watches, what is the largest square number that can occur? Largest cube?

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