Contest Quest Slides
Logic Labyrinth
Contest Mastery Challenge
Junior Math Quest
Use your arrow keys to navigate the chambers
Quest Sequence
1
The Challenge
A complex problem appears. Read it carefully and try to solve it on your own first.
2
The Secret
Stuck? The next slide reveals a "Labyrinth Secret"—a strategy hint to get you moving.
3
The Reveal
Finally, see the full solution and check your logic against ours.
Problem 01 Number Theory
A number \( N \) is between 50 and 100. It is a multiple of 6. If you add 1 to \( N \), the result is a multiple of 7.
What is the value of \( N \)?
Labyrinth Secret
"When you have two clues, start with the one that has fewer possibilities. List multiples of 6 first, then test them!"
Solution 01
90
1. Multiples of 6 between 50-100: 54, 60, 66, 72, 78, 84, 90, 96.
2. Add 1 to each: 55, 61, 67, 73, 79, 85, 91, 97.
3. 91 is a multiple of 7 (\( 7 \times 13 = 91 \)).
Problem 02 Geometry
A large rectangle is made by placing 5 identical squares side-by-side. The perimeter of one small square is 20 cm.
What is the perimeter of the large rectangle?
Labyrinth Secret
"Some sides 'disappear' when they touch! Sketch the shape and count only the outside edges."
Solution 02
60 cm
1. Square perimeter 20 \(\implies\) Side length = 5.
2. Width of rectangle = 5 squares \(\times\) 5 cm = 25 cm.
3. Height of rectangle = 5 cm.
4. Perimeter = \( 2 \times (25 + 5) = 60 \) cm.
Problem 03 Combinatorics
Four friends—Ava, Ben, Cho, and Dan—stand in line. If Ava must be at the very front, how many ways can they be arranged?
Labyrinth Secret
"When one spot is locked, ignore it! Just figure out how many ways to arrange the other 3 friends."
Solution 03
6 ways
1. Fixed spot: [Ava] [?] [?] [?]
2. Choices for 2nd spot: 3 people (B, C, or D).
3. Choices for 3rd spot: 2 people left.
4. Total = \( 3 \times 2 \times 1 = 6 \).
Problem 04 Algebra
At the Fruit Stand:
- 3 Apples and 1 Banana cost $1.60
- 1 Apple and 1 Banana cost $0.80
How much does one Apple cost?
Labyrinth Secret
"Look at what changes! Subtract the smaller bill from the larger one to see how much the extra apples cost."
Solution 04
$0.40
1. The difference between the lists is 2 Apples.
2. Difference in price: \( \$1.60 - \$0.80 = \$0.80 \).
3. If 2 Apples = $0.80, then 1 Apple = $0.40.
Problem 05 Number Theory
The sum of the digits of a 2-digit number is 11. If you reverse the digits, the new number is 9 greater than the original.
What is the original number?
Labyrinth Secret
"There aren't many pairs that add to 11! List them: (2,9), (3,8), (4,7), (5,6). Then just check which reversal works."
Solution 05
56
1. Pairs summing to 11: 29, 38, 47, 56.
2. Reverse of 56 is 65.
3. Difference: \( 65 - 56 = 9 \). Correct!
Problem 06 Geometry
A square has an area of 64 sq cm. If you double the length of each side, what is the area of the new square?
Labyrinth Secret
"Find the side length first! Then double it and find the new area. You'll notice the area grows faster than the sides."
Solution 06
256 cm²
1. Original side length = \(\sqrt{64} = 8\) cm.
2. Double the side length = \(8 \times 2 = 16\) cm.
3. New area = \(16 \times 16 = 256\) sq cm.
Problem 07 Combinatorics
How many different ways can you make 25 cents using only dimes and nickels?
Labyrinth Secret
"Be organized! Start with the most dimes possible (2) and work your way down to zero dimes."
Solution 07
3 ways
1. 2 Dimes, 1 Nickel.
2. 1 Dime, 3 Nickels.
3. 0 Dimes, 5 Nickels.
Problem 08 Algebra
Find the next number in this pattern:
1, 3, 7, 15, 31, ...
Labyrinth Secret
"Look at the jumps between the numbers! What happens to the difference each time?"
Solution 08
63
1. Rule: Double the number and add 1.
2. \( (31 \times 2) + 1 = 62 + 1 = 63 \).
3. Or jumps: +2, +4, +8, +16... next is +32.
Problem 09 Number Theory
What is the smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 2, 3, and 4?
Labyrinth Secret
"Find a number that 2, 3, and 4 all go into perfectly (LCM), then just add 1!"
Solution 09
13
1. Multiples of 4: 4, 8, 12...
2. 12 is divisible by 2, 3, and 4 (LCM).
3. Smallest number greater than 1 with rem 1: \( 12 + 1 = 13 \).
Problem 10 Geometry
A rectangle is 3 times as long as it is wide. If its area is 48 sq cm, what is its perimeter?
Labyrinth Secret
"Divide the area by 3 to find the area of a single square. Then find its side length."
Solution 10
32 cm
1. Area of 3 squares = 48 \(\implies\) 1 square = 16.
2. Side of square = 4 cm. Width = 4, Length = 12.
3. Perimeter = \( 2 \times (12 + 4) = 32 \) cm.
Problem 11 Combinatorics
If 5 people in a room each shake hands with every other person exactly once, how many handshakes occur?
Labyrinth Secret
"Person 1 shakes 4 hands. Person 2 shakes 3 *new* hands... keep going!"
Solution 11
10
1. Count: 4 + 3 + 2 + 1 = 10 handshakes.
2. (Person 5 has already shaken everyone's hand by then!)
Problem 12 Algebra
A mother is 4 times as old as her son. In 5 years, the sum of their ages will be 60. How old is the son now?
Labyrinth Secret
"Find their sum right now! If in 5 years it's 60, right now it is 10 years less (5 for each)."
Solution 12
10
1. Sum now = \( 60 - 10 = 50 \).
2. Son = \( \square \), Mother = \( 4 \times \square \). Total = \( 5 \times \square \).
3. \( 5 \times \square = 50 \implies \text{Son} = 10 \).
Problem 13 Number Theory
How many 2-digit numbers are there where the first digit is greater than the second digit?
Labyrinth Secret
"Be systematic! If the first digit is 1, what can the second be? If it's 2? 3?"
Solution 13
45
1. First digit 1: {10} (1 way)
2. First digit 2: {20, 21} (2 ways)
3. ... First digit 9: {90, 91...98} (9 ways)
4. Sum: \( 1+2+3+4+5+6+7+8+9 = 45 \).
Problem 14 Geometry
A rectangular box has dimensions 2 cm, 3 cm, and 4 cm. What is its total surface area?
Labyrinth Secret
"A box has 6 faces! There are three pairs of matching faces. Calculate the area of one of each type and double it."
Solution 14
52 cm²
1. Pair 1 (bottom/top): \( 2 \times 3 = 6 \). (x2 = 12)
2. Pair 2 (front/back): \( 3 \times 4 = 12 \). (x2 = 24)
3. Pair 3 (sides): \( 2 \times 4 = 8 \). (x2 = 16)
4. Total: \( 12 + 24 + 16 = 52 \) sq cm.
Problem 15 Combinatorics
How many different ways can you arrange the letters in the word LEVEL?
Labyrinth Secret
"Repeat letters make it tricky! If all letters were different, it would be 120 ways. Since we have two 'L's and two 'E's, we have to divide."
Solution 15
30
1. Total letters = 5. (Factorial \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)).
2. Divided by repeating L's (2) and E's (2).
3. \( 120 \div 2 \div 2 = 30 \).
Quest Complete
The Logic Labyrinth has been mastered!
4 Domains
15 Challenges
∞ Skill
1 Victory
Logic Labyrinth Worksheet
Logic Labyrinth
Junior Contest Prep Worksheet
Name: __________________________________
Date: ___________________________________
01
Number Theory
A number \( N \) is between 50 and 100. It is a multiple of 6. If you add 1 to \( N \), the result is a multiple of 7. What is \( N \)?
Work Area
Answer: _________________
02
Geometry
A large rectangle is made by placing 5 identical squares side-by-side. The perimeter of one small square is 20 cm. What is the perimeter of the large rectangle?
Work Area
Answer: _________________
03
Combinatorics
Four friends—Ava, Ben, Cho, and Dan—stand in line. If Ava must be at the very front, how many ways can the friends be arranged?
Work Area
Answer: _________________
04
Algebra
3 Apples and 1 Banana cost $1.60. 1 Apple and 1 Banana cost $0.80. How much does one Apple cost?
Work Area
Answer: _________________
05
Number Theory
The sum of digits of a 2-digit number is 11. Reversing the digits makes the number 9 greater than the original. What is the original number?
Work Area
Answer: _________________
06
Geometry
A square has an area of 64 sq cm. If you double the length of each side, what is the area of the new, larger square?
Work Area
Answer: _________________
07
Combinatorics
How many different ways can you make 25 cents using only dimes and nickels?
Work Area
Answer: _________________
08
Algebra
Find the next number in this pattern: 1, 3, 7, 15, 31, ...
Work Area
Answer: _________________
09
Number Theory
What is the smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 2, 3, and 4?
Work Area
Answer: _________________
10
Geometry
A rectangle is 3 times as long as it is wide. If its area is 48 sq cm, what is its perimeter?
Work Area
Answer: _________________
11
Combinatorics
If 5 people in a room each shake hands with every other person exactly once, how many handshakes occur?
