Sunday Times Teaser 3170 – Time to Admire the Adverts
by Danny Roth
Published Sunday June 25 2023 (link)
George and Martha use a local underground station, which has up and down escalators and an adjacent 168-step staircase. Martha always uses the escalator. George, being the fitness fanatic, always uses the stairs. In olden times, he could keep up with her while climbing or descending at seven steps per second. But now he can only ascend at N steps per second and descend at N + 1 steps per second, N being a whole number. They start together at the bottom and go up and down continually, losing negligible time in turning. Interestingly, at some point after they have both been all the way up and down, but before ten minutes have elapsed, Martha overtakes George precisely half-way up or half-way down.
After how many seconds does this happen?
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BRG permalink1234567891011121314151617181920# Martha's up and down times at the staircase mid-pointsmup = set(12 * n for n in range(1, 50, 4))mdn = set(12 * n for n in range(3, 50, 4))# Let George's uses of the staircase be numbered <n> from 1 upwards# with odd <n> for up and even <n> for down; his mid-point times on# the staircase are then 84.[n / N + (n - 1) / (N + 1)]# start with one full up/down cyclefor n in range(3, 50):# N < 7 because George is slower than Marthafor N in range(6, 0, -1):g = 84 * (n / N + (n - 1) / (N + 1))# times are less than ten minutesif g >= 600:breakif n % 2 and g in mup or not n % 2 and g in mdn:print(f"{g:.0f} seconds ({n = }, {N = }).")
