February 2017 Contest
1. On the planet Arret, recent games included tug-o-war contests between different species of animals. (Assume that each animal is of equal strength to every other animal of its species.)
If a contest between two snilas and three grofs ends in a draw, and
a contest between three ruttels and four snilas ends in a draw,
who would win between five grofs and two ruttels, or would it also end in a draw?
ANSWER: The Grofs win easily. Solved by Wes Vogler on 05 February 2017 at 19: 05.
2. A teacher writes a whole number less than 50,000 on the whiteboard. One student states that this number is a multiple of 2; a second student states that it is a multiple of 3; and so on until a twelfth student states that it is a multiple of 13. The teacher remarks that all of the students except two were correct, and that they (the ones who were wrong) spoke one right after the other.
What number did the teacher write on the whiteboard?
ANSWER: Let N be the number we are looking for. Since all the students, except two who spoke one after the other, were correct, in can be deduced that N can be divided by 1,2,3,4,5,6,10,11,12, and 13. This is because if 2 does not divide N, neither does 4; if 3 does not divide N, neither does 6; if 5 does not divide N, neither does 10; and so on. It is therefore necessary to examine two cases:
Case 1. N is not divisible by 8 and 9. In this case, however, the smallest number divisible by all the other numbers up to 13 (i.e., their least common multiple) is 60,060, a number greater than the one written on the whiteboard by the teacher.
Case 2. N is not divisible by 7 and 8. The least common multiple of the remaining numbers is 25,740, and since any other multiple of the same numbers is greater than 50,000, this must have been the number written by the teacher.
(Nobody solved this one. It was very tricky.)
3. If someone can dig a hole one foot square by one foot deep in one hour, at that rate how long would it take him to dig a hole eight feet square by eight feet deep?
ANSWER: 512 hours. One foot square by one foot deep = 1 x 1 x 1 = 1 cubit foot of dirt to remove, which takes one hour. Eight feet square by eight feet deep = 8 x 8 x 8 = 512 cubit feet of dirt to remove, which takes 512 hours at the same rate.
Solved by Wes Vogler on 06 February 2017 at 02: 07.
4. ‘How old are your three daughters? ‘ Fred asked a friend one day.
‘The product of their ages is 36’, replied his friend.
‘That’s not enough information, ‘ protested Fred.
‘The sum of their ages is the age of your oldest son.’
‘I still need more information.’
‘Very well. The oldest daughter, who is at least a year older than the other two, likes cats.’
‘Ah! Now I know their ages! ‘
What are the ages of the three girls?
ANSWER: The girls are 2,2, and 9 years old. Solved by Wes Vogler on 13 February 2017 at 15: 26.
Since the product of the ages of the three sisters is 36, the sisters’ ages will be one of the following sets of three numbers: (1,1,36) : (1,2,18) : (1,3,12) : (1,4,9) : (1,6,6) : (2,2,9) : (2,3,6) : or (3,3,4) .
These sets of numbers add up to, respectively,38,21,16,14,13,13,11, and 10. Since Fred knows the age of his own oldest son, that’s all the information he needs, UNLESS his son is 13, in which case the ages could have been either (1,6,6) or (2,2,9) .
That’s why he said he still needed more information, which, when provided, enabled Fred to reach the right conclusion.
5. According to Poem Hunter’s ‘popularity’ list, who is the number one poet in Thailand right now?
ANSWER: Darwin Henry Beuning
Solved by Wes Vogler on 05 February 2017 at 18: 37
(My friend Hank Beuning is living in Thailand for the next couple of months. Hank is the one who introduced me to Poem Hunter and got me writing again.)
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