UOJMC Training Protocol

Year 9 Algorithmic and Heuristic Problem Solving

Designed by Mr.Shen

Abstract Number Theory & Logic

Training Session: Number Theory Fundamentals

1. Using a Calculator to Find Terminal Digit Patterns

You cannot calculate massive exponents like $8^{2027}$ on a standard calculator. Instead, use your calculator to find the first few powers and identify the repeating pattern of the last digits.

  • Calculate $8^1 = 8$, $8^2 = 64$, $8^3 = 512$, $8^4 = 4096$, $8^5 = 32768$.
  • Notice the last digits: 8, 4, 2, 6. The pattern repeats every 4 powers.
  • Divide the exponent by the cycle length (e.g., $2027 \div 4$). The remainder tells you the position in the cycle! (Remainder 3 = 3rd number in pattern).

2. Index Rules with the Same Base

To divide or multiply exponents, the bases must be identical. Use prime factorization to convert bases before applying the index rules.

  • Rule: $\frac{a^m}{a^n} = a^{m-n}$
  • Example: To simplify $\frac{3^{10}}{27^2}$, first convert $27$ to base $3$. Since $27 = 3^3$, the denominator becomes $27^2 = (3^3)^2 = 3^6$.
  • Now apply the rule: $\frac{3^{10}}{3^6} = 3^4$.

1. Determine the exact terminal (last) digit of the exponential expression $8^{2027}$.

2. Simplify the fractional expression $\frac{3^{10}}{27^2}$ to its fundamental integer value. Ensure base variables are equivalent prior to exponent manipulation.

=

Algorithmic Resolution:

Q1 Step-by-Step:
  1. Identify the repeating pattern of the last digit for powers of 8: $8^1=8$, $8^2=64 \rightarrow 4$, $8^3=512 \rightarrow 2$, $8^4=4096 \rightarrow 6$. The cycle is 8, 4, 2, 6 (length of 4).
  2. Divide the exponent by the cycle length: $2027 \div 4 = 506$ with a remainder of 3.
  3. The remainder tells us the position in the cycle. The 3rd number in the sequence [8, 4, 2, 6] is 2. (Answer: A)
Q2 Step-by-Step:
  1. Harmonize the bases. We know that $27$ can be written as $3^3$.
  2. Substitute this into the denominator: $27^2 = (3^3)^2$.
  3. Apply the power of a power rule (multiply exponents): $(3^3)^2 = 3^6$.
  4. Rewrite the original fraction with the new denominator: $\frac{3^{10}}{3^6}$.
  5. Apply the quotient rule (subtract exponents): $3^{10-6} = 3^4$.
  6. Calculate the final integer: $3 \times 3 \times 3 \times 3 = 81$.

Spatial Geometry & Area Calculation

Training Session: Step-by-Step Spatial Abstraction

Part 1: The Painted Cube Challenge

Imagine you have a large 3x3x3 Rubik's Cube made of 27 smaller wooden blocks. You dip the entire large cube into a bucket of black paint. Then, you break it apart. How many of the small inner blocks have absolutely NO paint on them? Use the interactive model below to test your thinking.

Core
Step-by-Step Thinking:
  1. Visualize the Paint: The paint only touches the outside layer (the "skin") of the large cube. The center is completely protected.
  2. Peel the Skin: To find the unpainted core, we must mentally "peel off" the outer layer from every direction.
  3. Subtracting from Dimensions:
    • Peel the top and bottom = subtract 2 from the height.
    • Peel the left and right = subtract 2 from the width.
    • Peel the front and back = subtract 2 from the depth.
  4. The Formula: For a cube that is $n \times n \times n$, the unpainted inner core is always $(n-2) \times (n-2) \times (n-2)$.

Part 2: Cracking the 30-60-90 Triangle

Some right-angled triangles have special angles ($30^\circ$, $60^\circ$, and $90^\circ$). You don't need to do complex math if you know their secret ratio! Follow the interactive graph below to understand how this ratio is derived using Pythagoras.

60° 60° 60° 30° 2 2 2 1 1 h =? √3
The Origin: Start with a perfect equilateral triangle where every side is exactly 2, and every angle is $60^\circ$.
The Slice: Cut it straight down the middle. You now have a right-angled triangle. The bottom base was cut in half, so it's exactly 1 (this is the shortest side, opposite the $30^\circ$ angle). The slanted hypotenuse hasn't changed; it's still 2.
The Middle Side: Using Pythagoras ($a^2 + b^2 = c^2$), we need to find the height $h$. We plug in the knowns: $1^2 + h^2 = 2^2$.
The Golden Rule: $1 + h^2 = 4$, which means $h^2 = 3$, so the height is exactly $\sqrt{3}$. The ratio of ANY 30-60-90 triangle is ALWAYS $1 : \sqrt{3} : 2$. If the shortest side is $x$, the hypotenuse is $2x$, and the height is $x\sqrt{3}$.

1. A solid $5 \times 5 \times 5$ cubic volume is painted black on its exterior surface. It is then sliced into $125$ unit cubes ($1 \times 1 \times 1$). How many unit cubes will possess exactly zero black faces (representing the unpainted interior)?

unit cubes

2. Within a right-angled triangle containing a $30^\circ$ angle, if the shortest side length (opposite the $30^\circ$ angle) is 3 cm, what is the exact area of the entire triangle?

Algorithmic Resolution:

Q1 Step-by-Step:
  1. Understand the formula for the unpainted inner core of a cube: $(n-2) \times (n-2) \times (n-2)$.
  2. Identify the dimension of the large cube: It is a $5 \times 5 \times 5$ cube, so $n = 5$.
  3. Subtract 2 from each dimension to "peel off" the painted outer layer: $5 - 2 = 3$.
  4. Calculate the volume of this new $3 \times 3 \times 3$ inner core: $3 \times 3 \times 3 = 27$.
Q2 Step-by-Step:
  1. Recall the 30-60-90 triangle ratio: The sides are in the ratio $1 : \sqrt{3} : 2$.
  2. Identify the given side: The shortest side (opposite the $30^\circ$ angle) is the base, $b = 3$.
  3. Find the height: The height (opposite the $60^\circ$ angle) is the short side multiplied by $\sqrt{3}$. So, $h = 3\sqrt{3}$.
  4. Apply the area formula for a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
  5. Substitute the values: $\text{Area} = \frac{1}{2} \times 3 \times 3\sqrt{3} = \frac{9\sqrt{3}}{2}$. (Answer: B)

Applied Physics & Rates

Training Session: Step-by-Step Backward Induction

1. Reversing Exponential Decay (Half-Life)

A "half-life" is the amount of time it takes for a radioactive substance to shrink exactly in half. But what if a question gives you the FINAL amount and asks what you started with? You must think backwards like a detective. Here is a visual representation of how a 160g substance decays over 3-minute intervals:

160g
Start
Halve 3 mins
80g
1st
Half-Life
Halve 3 mins
40g
2nd
Half-Life
Halve 3 mins
20g
3rd
Half-Life
Step-by-Step Thinking (Reversing the Arrows):
  1. Count the Intervals: Divide the total time by the half-life time. (e.g., If 9 minutes passed, and a half-life is 3 minutes... $9 \div 3 = 3$ halving events).
  2. Reverse the Action: Since time is going backwards (moving from right to left on the diagram), we don't halve the material, we DOUBLE it.
  3. Execute: Take the final weight, and multiply it by 2 for every event. (e.g., $20\text{g} \times 2 \times 2 \times 2 = 160\text{g}$).

2. Algebraic Substitution in Physics

You will often be given strange formulas from physics, like $p = vi$ (Power = Voltage $\times$ Current). Don't panic! You don't need to be a physicist; you just need to be a puzzle solver.

Step-by-Step Thinking:
  1. Treat letters as empty boxes: Write down the formula exactly as given: $p = v \times i$.
  2. Plug in the numbers: The question will give you two of the three numbers. Put them into their matching boxes. (e.g., If Power is 120 and Current is 8, write $120 = v \times 8$).
  3. Isolate the unknown: Use basic algebra to get the letter by itself. To undo "multiply by 8", you divide both sides by 8. ($120 \div 8 = v$).

1. A radioactive isotope possesses a half-life of exactly 3 minutes. An unknown quantity is isolated. After an elapsed time of 15 minutes, 4g of material remains. Calculate the initial mass in grams.

grams

2. Electrical power $p$ (in watts) is calculated via the equation $p = vi$, where $v$ is voltage and $i$ is current (in amperes). If a specific circuit demands a fixed power output of 120W and the current is strictly limited to 8A, calculate the required voltage.

Algorithmic Resolution:

Q1 Step-by-Step:
  1. Determine the number of half-life intervals that passed: $15 \text{ minutes total} \div 3 \text{ minutes per half-life} = 5 \text{ intervals}$.
  2. Start with the final remaining mass: $4\text{g}$.
  3. Apply backward induction by doubling the mass for each of the 5 intervals:
    • 1st reversal: $4 \times 2 = 8\text{g}$
    • 2nd reversal: $8 \times 2 = 16\text{g}$
    • 3rd reversal: $16 \times 2 = 32\text{g}$
    • 4th reversal: $32 \times 2 = 64\text{g}$
    • 5th reversal: $64 \times 2 = 128\text{g}$
  4. The initial mass was $128\text{g}$.
Q2 Step-by-Step:
  1. Write down the formula provided: $p = v \times i$.
  2. Substitute the known values into the equation: Power ($p$) is $120$, Current ($i$) is $8$. The equation becomes $120 = v \times 8$.
  3. Isolate the unknown variable ($v$) by using the inverse operation of multiplication. Divide both sides by $8$.
  4. Calculate the result: $120 \div 8 = 15$. The required voltage is $15\text{V}$. (Answer: C)

Statistics & Set Theory

Training Session: Step-by-Step Set Logic & Sums

1. Hacking the "Mean Shift" Puzzle

Competitions love to say: "The average score of 10 people is 50. A new person joins, and the average becomes 51. What did the new person score?" Never try to guess individual scores. Instead, always look for the hidden total sum.

Step-by-Step Thinking:
  1. The Magic Formula: $\text{Total Sum} = \text{Average (Mean)} \times \text{Number of People}$.
  2. Find the Old Sum: Calculate the total points of the original group. (e.g., $10 \text{ people} \times 50 \text{ points} = 500 \text{ total points}$).
  3. Find the New Sum: Calculate the total points of the new, larger group. (e.g., $11 \text{ people} \times 51 \text{ points} = 561 \text{ total points}$).
  4. Find the Difference: Subtract the old sum from the new sum. That difference is exactly what the new person scored! ($561 - 500 = 61$).

2. Diagrammatic Translation (Venn Diagrams)

When a problem talks about populations with overlapping interests (e.g., students who play BOTH soccer and basketball), it is a trap designed to make you count the same people twice. Draw a picture to save yourself.

Total Soccer (240) Total Basketball (320) 160 80 240 Only Soccer Play Both Only B-Ball
Step-by-Step Thinking:
  1. Find the Overlap First: The most important rule of Venn diagrams is to ALWAYS fill in the middle intersection first. If 80 people play both, put 80 in the middle.
  2. Subtract for the "Only" Section: If 240 people play Soccer in total, and 80 are already accounted for in the middle, then the people who play ONLY Soccer is $240 - 80 = \mathbf{160}$.
  3. Why? If you just added 240 + 320 to find out how many people play sports, you would be counting those 80 overlapping athletes twice! Subtracting ensures each person is counted exactly once.

1. The arithmetic mean score of a cohort of 15 students is 60. A 16th student submits a late examination, which alters the cohort's mean score to exactly 62. Deduce the precise score achieved by the 16th student.

points

2. From a survey pool of 800 students, 30% play soccer and 40% play basketball. If precisely one-quarter (25%) of the students who play basketball also play soccer, how many students play soccer exclusively (but not basketball)?

Algorithmic Resolution:

Q1 Step-by-Step:
  1. Calculate the total sum of points for the original cohort: $15 \text{ students} \times 60 \text{ points (mean)} = 900 \text{ total points}$.
  2. Calculate the new total sum of points for the expanded cohort: $16 \text{ students} \times 62 \text{ points (new mean)} = 992 \text{ total points}$.
  3. Find the difference between the two sums to isolate the 16th student's exact score: $992 - 900 = 92$.
Q2 Step-by-Step:
  1. Find the total number of students who play soccer: $800 \times 0.30 = 240 \text{ soccer players}$.
  2. Find the total number of students who play basketball: $800 \times 0.40 = 320 \text{ basketball players}$.
  3. Calculate the intersection (students who play BOTH). The problem states this is 25% of the basketball players: $320 \times 0.25 = 80 \text{ students play both}$.
  4. To find the students who play only soccer, take the total soccer players and subtract the intersecting group: $240 - 80 = 160 \text{ exclusive soccer players}$. (Answer: B)