1. A three digit number has digits in descending
order (first digit is greater than the 2nd, which is
greater than the third). Each digit is different. It is
a multiple of 15. It is not a multiple of 2. No digit
is a 9. What is the number?
2. A three digit number has digits in ascending order
(first digit is less than the 2nd, which is less than
the third). Each digit is different. It is a multiple
of 15. It is less than 300. What is the number?
3. The product of 124 011 and 452 008 is divided by
the sum of 251 and 749. What is the remainder?
4. The product of 24 209 and 37 103 is divided by the
sum of 17 and 83. What is the remainder?
5. Doris, Eggbert and Fergus are playing Chink Splat
Ball. Doris scored 3 Chinks and a 2 Splats for a
total score of 21. Eggbert scored 5 Chinks and 7
Splats for a score of 46. Fergus scored 4 Chinks
and 6 Splats. What is Fergus’ total score?
6. Gertrude, Harold and Irene are playing Chink Splat
Ball. Gertrude scored 7 Chinks and 2 Splats for a
total score of 31. Harold scored 2 Chinks and 3
Splats for a score of 21. Irene scored 4 Chinks and
6 Splats. What is Irene’s total score?
7. Jane was arranging more than 60 cans. She can
arrange them in a triangle, or a square pyramid.
What is the least number of cans she could have?
8. Sian was arranging cans (note ‘cans’ is a plural –
so more than 1 can). He can arrange them in a
triangle or a square pyramid. What is the least
number of cans he could have?
9. How many counting numbers less than 100 have
exactly 3 different factors? (Yes, 1 is a factor of
every counting number.)
10. What is the smallest positive number with exactly
5 different factors? (Yes, 1 is a factor of every
counting number.)
11. Habib’s Science Fair project compared the output
of different styles of wind turbines.
6 turbines of style A produced 6 KWh in 6 hours.
3 turbines of style B produced 3 KWh in 3 hours.
2 turbines of style C produced 2KWh in 2 hours.
1 turbine of style D produced 1KWh in half an
hour. Assuming constant wind, if Habib had one
of each style running for 12 hours, how much
electricity would he produce?
12. Ali’s Science Fair project compared the output of
different styles of wind turbines.
5 turbines of style A produced 5 KWh in 6 hours.
3 turbines of style B produced 3 KWh in 3 hours.
2 turbines of style C produced 4KWh in 2 hours.
1 turbine of style D produced 1KWh in half an
hour. Assuming constant wind, if Ali had one of
each style running for 15 hours, how much
electricity would he produce?
13. Australians show date as dd/mm. Americans
show it as mm/dd. If Ozzy writes the date of a
party in Australian format in an e-mail to Hank, an
American friend, how many days of the year could
Hank think it is a different date? (i.e. don’t count
dates like 28/2)
14. A digital clock displays the time as HH:MM. From
1:50am to 10:50 am, how many times is MM less
than HH?
15. Karen and Sharon are sisters with the same
birthday. In 2000 Karen was four times as old as
Sharon. In 2002 Karen was three times as old as
Sharon. In what year will the sum of their ages be
60?
16. Jo and Flo are sisters with the same birthday. In
2000 Jo was three times as old as Flo. In 2003 Jo
was twice as old as Flo. In what year will the sum
of their ages be 50?
17. Hanne and Polema were running around a track in
opposite directions. They pass each other every
90 seconds. Hanne takes 2 minutes to run 1 lap.
How many minutes does Polema take to run a
lap?
18. Hanne and Polema were running around a track in
opposite directions. They pass each other every
80 seconds. Hanne takes 2 minutes to run 1 lap.
How many minutes does Polema take to run a
lap?
19. Sally was arranging cans. “If only I had 4 more, I
could arrange them in a square,” Sally lamented.
“That’s OK,” said Mae, “you have the exact
number for a triangular pyramid.” What is the
smallest number of cans Sally could have?
20. Sally was arranging cans (plural, so more than 1).
“If only I had 8 more, I could arrange them in a
square,” Sally lamented. “That’s OK,” said Mae,
“you have the exact number for a triangular
pyramid.” What is the smallest number of cans
Sally could have?
21. Frank had 6 white socks and 4 black socks in a
drawer. If he pulls out 2 at random, what is the
chance they are a pair?
22. Ben has 6 white socks and 2 black socks in a
drawer. If he pulls out 2 at random, what is the
chance they are a pair?
Relay Questions – Answers
Only give out questions in order. Only give odd questions to odd group, and even questions to even group.
Teams may not return to questions they have passed.
Make sure teams can’t see these answers.
Every correct answer is 1 point, even if on third attempt.
Try 1
Try 2
Try 3
Q1 answer 765
Try 1
Try 2
Try 2
Try 2
Try 3
Try 1
Try 2
Try 3
Try 1
Try 2
Q10 answer 16
Try 1
Try 1
Try 3
Q13 answer 132
Try 1
Try 2
Try 2
Try 3
Try 1
Try 2
Try 3
Try 3
Try 3
Try 1
Try 1
Q21 answer 7/15
Try 2
Q22 answer 4/7
Try 1
Try 2
Try 1
Try 2
Try 1
Try 2
Try 3
Try 1
Try 2
Q19 answer 165
Try 1
Try 2
Try 3
Q4 answer 27
Try 3
Try 1
Try 2
Try 3
Q8 answer 55
Try 3
Try 1
Try 2
Try 3
Q12 answer 52.5 KWh
Try 3
Q15 answer 2020
Q18 answer 4 minutes
Try 3
Try 3
Q11 answer 36 KWh
Q17 answer 6 minutes
Try 2
Try 2
Q7 answer 91
Q14 answer 45
Try 3
Try 1
Q3 answer 88
Q6 answer 42
Q9 answer 4 (4,9,25,49)
Try 2
Try 3
Q2 answer 135
Q5 answer 38
Try 1
Try 1
Try 1
Try 2
Try 3
Q16 answer 2019
Try 3
Try 1
Try 2
Q20 answer 56
Try 3
Relay Questions Worked Solutions
1.
A three digit number has digits in descending order (first digit is greater than the 2nd, which is greater than the
third). Each digit is different. It is a multiple of 15. It is not a multiple of 2. No digit is a 9. What is the number?
Multiple of 15 => multiple of 3 and multiple of 5.
Multiple of 5, and not multiple of 2 => ends in a 5.
No digit a 9 => 765, 875, or 865.
Multiple of 3 => digit sum is a multiple of 3 => 765
2.
A three digit number has digits in ascending order (first digit is less than the 2nd, which is less than the third).
Each digit is different. It is a multiple of 15. It is less than 300. What is the number?
Multiple of 15 => multiple of 5 and multiple of 3
Multiple of 5 => ends in a 5 or a 0
Ascending => can’t end in 0, must end in 5
Less than 300 => first digit 1 or 2
=> 125, 135, 145, 235, 245
Multiple of 3 => digit sum is a multiple of 3 => 145
3.
The product of 124 011 and 452 008 is divided by the sum of 251 and 749. What is the remainder?
251 + 749 = 1000
Let a = 124, b = 452
124 011 x 452 008 = (1000a + 11)x(1000b + 8)
= 1000x1000ab + 1000x8a + 1000x11b + 11x8
= 1000 x (1000ab + 8a + 11b) + 88
= some multiple of 1000 that we don’t care about + 88
Remainder is 88
4.
The product of 24 209 and 37 103 is divided by the sum of 17 and 83. What is the remainder?
17 + 83 = 100
Let a = 242, b = 371
24 209 x 37 103 = (100a + 9) x (100b + 3)
= 100 x 100 ab + 100 x 3a + 100 x 9b + 9 x 3
= 100 (100ab + 3a + 9b) + 27
= some multiple of 100 we don’t care about + 27
Remainder is 27
5.
Doris, Eggbert and Fergus are playing Chink Splat Ball. Doris scored 3 Chinks and a 2 Splats for a total score of
21. Eggbert scored 5 Chinks and 7 Splats for a score of 46. Fergus scored 4 Chinks and 6 Splats. What is Fergus’ total
score?
Use guess and check, or let c be number of chinks, and s be number of splats
3c +2s = 21
from Doris. Multiple by 5.
15c + 10s = 105 (2)
5c + 7s = 46
from Eggbert. Multiply by 3.
15c + 21s = 138
take away previous equation (2)
11s = 33
s=3
c=5
Fergus scored 4c + 6s = 20 + 18 = 38
6.
Gertrude, Harold and Irene are playing Chink Splat Ball. Gertrude scored 7 Chinks and 2 Splats for a total score
of 31. Harold scored 2 Chinks and 3 Splats for a score of 21. Irene scored 4 Chinks and 6 Splats. What is Irene’s total
score?
Use guess and check or let c = number of chinks, s = number of splats
7c + 2s = 31
From Gertrude. Multiply by 3
21c + 6s = 93
(2)
2c + 3s = 21
From Harold. Multiply by 2
4c + 6s = 42
(2) take away this equation
17c = 51
c = 3, so s = 5
Irene scored 4c + 6s = 12 + 30 = 42
7.
Jane was arranging more than 60 cans. She can arrange them in a triangle, or a square pyramid. What is the
least number of cans she could have?
Triangle numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, …
Square pyramid numbers 1, 5, 14, 30, 55, 91
91 is the first number common to both lists, after 60.
8.
Sian was arranging cans (note ‘cans’ is a plural – so more than 1 can). He can arrange them in a triangle or a
square pyramid. What is the least number of cans he could have?
55 is the first number common to both lists, after 1.
9.
How many counting numbers less than 100 have exactly 3 different factors? (Yes, 1 is a factor of every counting
number.)
Squares of primes have exactly three factors. E.g. 4 has 1, 2 and 4. 1 is not a prime, and 1 has exactly 1 factor.
Suitable numbers are 4, 9, 25 and 49. So answer is 4.
10.
What is the smallest positive number with exactly 5 different factors? (Yes, 1 is a factor of every counting
number.)
Number 1
2
Factors 1
2
16 is the first one
11.
3
2
4
3
5
2
6
4
7
2
8
4
9
3
10
4
11
2
12
6
13
2
14
4
15
4
16
5
Habib’s Science Fair project compared the output of different styles of wind turbines.
6 turbines of style A produced 6 KWh in 6 hours. 3 turbines of style B produced 3 KWh in 3 hours.
2 turbines of style C produced 2KWh in 2 hours.
1 turbine of style D produced 1KWh in half an hour. Assuming constant wind, if Habib had one of each style running for
12 hours, how much electricity would he produce?
In 12 hours turbine A would produce 6KWh / 6 turbines x 2 (lots of 6 hours in 12 hours)
turbine B would produce 3 KWh / 3 turbines x 4 (lots of 3 hours in 12 hours)
turbine C would produce 2KWh / 2 turbines x 6
turbine D 1KWh x 24 (which is how many half hours in 12 hours)
Total is 36KWh
= 2KWh
= 4KWh
= 6KWh
= 24KWh
12.
Ali’s Science Fair project compared the output of different styles of wind turbines.
5 turbines of style A produced 5 KWh in 6 hours. 3 turbines of style B produced 3 KWh in 3 hours.
2 turbines of style C produced 4KWh in 2 hours.
1 turbine of style D produced 1KWh in half an hour. Assuming constant wind, if Ali had one of each style running for 15
hours, how much electricity would he produce?
In one hour turbine A would produce 5KWh / 5 turbines /6 = 1/6
Turbine B 3 / 3 / 3 = 1/3
Turbine C 4 / 2 / 2 = 1
Turbine D 1 / 0,5 = 2
So a total of 3.5 KWh in 1 hour, so 52.5 KWh in 15 hours
13.
Australians show date as dd/mm. Americans show it as mm/dd. If Ozzy writes the date of a party in Australian
format in an e-mail to Hank, an American friend, how many days of the year could Hank think it is a different date? (i.e.
don’t count dates like 28/2)
Dates can only be legal backwards if the day of the month is 1..12. Dates aren’t confusing if the day and month are the
same (e.g. 3/3/14). So 11 days every month are confusing. 11 x 12 = 132.
14.
A digital clock displays the time as HH:MM. From 1:50am to 10:50 am, how many times is MM less than HH?
2:01, 3:01, 3:02, 4:01, 4:02, 4:03, …10:09.
So 1 + 2 + 3 + … + 9 = 45
15.
Karen and Sharon are sisters with the same birthday. In 2000 Karen was four times as old as Sharon. In 2002
Karen was three times as old as Sharon. In what year will the sum of their ages be 60?
Can use guess and check or let s be Sharon’s age in 2000.
In 2000, Karen was 4s.
In 2002
4s+2 = 3(s+2)
4s + 2 = 3s +6
s = 4.
So in 2000 they were 16 and 4, which has a sum of 20. Each year the sum of their ages increases by 2. So need (6020)/2 more years, 20 more years. In 2020 the sum of their ages will be 60.
16.
Jo and Flo are sisters with the same birthday. In 2000 Jo was three times as old as Flo. In 2003 Jo was twice as
old as Flo. In what year will the sum of their ages be 50?
Can use guess and check, or let f be Flo’s age in 2000. So Jo is 3f in 2000.
In 2003
3f + 3 = 2(f + 3)
= 2f + 6
So f = 3. So in 2000 Flo was 3, and Jo was 9. Sum of ages was 12. Every year their sum increases by 2, (50-12)/2 years to
go. 19 years later, in 2019 the sum of their ages will be 50.
17.
Hanne and Polema were running around a track in opposite directions. They pass each other every 90 seconds.
Hanne takes 2 minutes to run 1 lap. How many minutes does Polema take to run a lap?
Combined they run a whole lap in 90 seconds. In that time, Hanne has run ¾ of a lap, so Polema must have run ¼ of a
lap. This means Polema is running at 1/3 of Hanne’s speed, so she will take 3 times as long to run a lap.
Polema takes 6 minutes to run 1 lap.
18.
Hanne and Polema were running around a track in opposite directions. They pass each other every 80 seconds.
Hanne takes 2 minutes to run 1 lap. How many minutes does Polema take to run a lap?
In 80 seconds, Hanne has run 80/120 = 2/3 of a lap, so Polema must have run the other 1/3. Polema is half as fast as
Hanne, so took 4 minutes to run a lap.
19.
Sally was arranging cans. “If only I had 4 more, I could arrange them in a square,” Sally lamented. “That’s OK,”
said Mae, “you have the exact number for a triangular pyramid.” What is the smallest number of cans Sally could have?
Number
1 2 3 4 5 6 7
8
9 10 11 12 13
Square
1 4 9 16 25 36 49 64 81 100 121 144 169
Triangular 1 4 10 20 35 56 84 120 165
Pyramid
Select the first triangular pyramid number that is 4 less than a square. So 165.
20.
Sally was arranging cans (plural, so more than 1). “If only I had 8 more, I could arrange them in a square,” Sally
lamented. “That’s OK,” said Mae, “you have the exact number for a triangular pyramid.” What is the smallest number
of cans Sally could have?
56 is the first triangular pyramid number, greater than 1, that is 8 less than a square.
21.
pair?
Frank had 6 white socks and 4 black socks in a drawer. If he pulls out 2 at random, what is the chance they are a
Chance of white then white = 6/10 x 5 / 9
= 30/90
Chance of black then black = 4/10 x 3 / 9
= 12 /90
Total chance = 30/90 + 12/90
= 42/90
= 7 / 15
22.
pair?
Ben has 6 white socks and 2 black socks in a drawer. If he pulls out 2 at random, what is the chance they are a
Chance of white then white = 6/8 x 5/7
= 30/56
Chance of black then black = 2/8 x 1/7
= 2 / 56
Total chance = 30/56 + 2/56
= 32/56
= 4/7