One Martian, one Venusian and one Human reside on Pluto.
One day they make the following conversation:
Martian: I have spent 1/12 of my life in Pluto.
Human: I also have.
Venusian: Me too.
Martian: But Venusian and I have spent much more time here than you, Human.
Human: However Venusian and I are of the same age.
Venusian: I have lived 300 Earth years.
Martian: Venusian and I have been on Pluto for the last 13 years.
It is known that Human and Martian together have lived for 104 Earth years.
Find their respective ages.
HINT:
The numbers are not necessarily expressed in base 10 .
Source: 1971 IMO longlisted problem.
Assuming the Human and the narrator quoting 104 Earth years are both using base 10 and that the Martian and Venusian are each consistently using the same base in different locutions but not necessarily the same base as each other:
Depending on the base that the Martian uses, his years on Pluto and his age could be (all figures in decimal):
base years on Pluto age Human's age 1/12 of Human's
life
4 7 42 62
5 8 56 48 4
6 9 72 32
7 10 90 14
8 11 110
9 12
10 13
11 14
12 15
This is also assuming that when the Human is specifying 1/12 of his life, it implies his integral age is a multiple of 12.
If the line with the 48-year-old human is correct then the Venusian is also 48 and must be using base 4 to quote his age as 300.
Is it true that Venusian and Martian have spent much more time on Pluto than Human? The Martian has spent 8 years to Human's 4. Venusian has spent 1/6 of his 48 years on Pluto--that's 8 years also; each has spent twice as many years on Pluto as has the Human.
The Human and the Venusian are each 48 Earth years old and the Martian is 56.
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Posted by Charlie
on 2018-06-19 11:23:38 |
tabular approach (spoiler)
