2.2 Mathematical Logic, Cryptarithms & Balance Puzzles

2.2 Mathematical Logic, Cryptarithms & Balance Puzzles

The Architecture of Non-Routine Mathematical Logic

Unlike standardized computational problems where an algorithm leads directly from inputs to an answer, non-routine logic puzzles present an underdetermined or masked system of constraints. Candidates must locate hidden invariants—properties that remain constant regardless of configuration—and systematically reduce possibilities.

Four key principles guide non-routine puzzle analysis:

1. The Invariant Principle: Identify quantities that cannot change (e.g., total weight on a balanced beam, total sum of digits, magic square row totals).
2. Equivalence and Substitution: Replace multi-element collections with equivalent smaller units.
3. Extreme Value and Boundary Analysis: Examine minimum and maximum values to limit search spaces.
4. Parity Constraints: Use the arithmetic of even and odd numbers to rule out impossible parity combinations.

Balance Scale Logic and Weight Substitution

Balance scale problems present two or three scales in equilibrium and ask for the equivalence of an unknown combination. Each scale represents a balanced equation ($LHS = RHS$).

Principles of Balance Scale Manipulation

• Elimination of Common Elements: If an identical shape appears on both pans of a balanced scale, removing it preserves balance ($A + B = A + C \implies B = C$).
• Direct Substitution: If Scale 1 proves that 1 Sphere balances 2 Cubes, every Sphere appearing on any other scale can be replaced by 2 Cubes.
• Additive Scaling: If 1 Cylinder balances 3 Cones, then 2 Cylinders balance 6 Cones.

Worked Example: Multi-Scale Weight Equivalence

Scenario: Three balance scales display geometric solids in perfect balance.

• Scale 1: 1 Cylinder and 1 Cone balance 1 Sphere.
• Scale 2: 1 Sphere balances 1 Cone and 4 Cubes.
• Scale 3: 1 Cylinder balances 2 Cubes and 1 Cone.

Question: How many Cubes balance 1 Cylinder?

Let us solve this systematically through step-by-step substitution:

Therefore, exactly 4 Cubes balance 1 Cylinder. Notice how eliminating the common Cone from both pans immediately produced the exact answer without needing trial and error.

Cryptarithms and Alphametic Puzzles

A cryptarithm (or alphametic puzzle) is a mathematical computation where digits are replaced by letters of the alphabet.

Fundamental Rules of Cryptarithms

1. Each letter represents a distinct digit from $0$ to $9$. Two different letters can never share the same digit.
2. The leading digit of a multi-digit number cannot be zero (e.g., in a three-digit number $ABC$, $A \neq 0$).
3. The standard rules of base-10 place-value arithmetic apply throughout.

Key Deductive Rules for Addition Cryptarithms

• The Leading Carry Rule: When two $k$-digit numbers are added to produce a $(k+1)$-digit sum, the leading digit of the sum must be 1. In base 10, the maximum sum of two single digits plus a carry is $9 + 9 + 1 = 19$, so a carry out into a new column can never exceed 1.
• Zero Identification: If $X + Y = X$ in a column (with no carry), then $Y = 0$. If there is a carry into the next column, $Y = 9$ with an incoming carry of 1.
• Parity Analysis: Adding two identical digits always generates an even units digit: $D + D = \dots (\text{even})$, unless an incoming carry adds 1, making it odd.

Step-by-Step Cryptarithm Solution

Problem: Solve the addition cryptarithm where each letter represents a distinct digit from 1 to 9: $\begin{array}{rl} & A\ B \\ + & B\ A \\ \hline & C\ D\ C \end{array}$

Step 1: Apply the Leading Carry Rule to find $C$. The sum of two 2-digit numbers ($AB + BA$) produces a 3-digit number ($CDC$). The maximum sum of two 2-digit numbers is $99 + 99 = 198$. Thus, the hundreds digit $C$ must equal 1.

Step 2: Express the numbers using base-10 place value.

• $AB = 10A + B$
• $BA = 10B + A$
• $CDC = 100C + 10D + C = 100(1) + 10D + 1 = 101 + 10D$

Summing the left-hand side gives: $(10A + B) + (10B + A) = 11A + 11B = 11(A + B)$

Step 3: Analyze divisibility by 11. Since $11(A + B) = 101 + 10D$, the 3-digit number $1D1$ must be a multiple of 11. Using the divisibility rule for 11 (alternating sum of digits: $(1 + 1) - D = 2 - D$), we see that $2 - D = 0 \implies D = 2$.

Thus, $CDC = 121$. Dividing by 11 gives $A + B = 11$. Since $A$ and $B$ must be distinct single digits not equal to 1 or 2, valid pairs could be $(3, 8), (4, 7), (5, 6)$.

Matrix Number Puzzles & Magic Squares

A classic magic square is an $n \times n$ grid filled with distinct positive integers where each row, column, and main diagonal sums to the identical constant, known as the Magic Sum ($S$).

The $3 \times 3$ Magic Square Properties

For a standard $3 \times 3$ grid using numbers $1$ through $9$:

1. Total Sum: $\sum_{i=1}^{9} i = \frac{9 \times 10}{2} = 45$.
2. Magic Sum: Since there are 3 rows summing to 45, each row, column, and diagonal must equal $S = \frac{45}{3} = 15$.
3. The Center Cell Rule: The center cell is intersected by 4 lines (1 horizontal, 1 vertical, 2 diagonals). Mathematically, the center cell must equal exactly one-third of the Magic Sum: $\text{Center Cell} = \frac{S}{3} = \frac{15}{3} = 5$

Operational Rule-Making (Defined Operators)

Synthetic operator puzzles introduce non-standard symbols (such as $\star, \odot, \Delta, \diamondsuit$) defined by a specific mathematical instruction. These test algebraic substitution and order of operations.

Structure of Defined Operators

Given an operation definition: $p \star q = 3p - 2q + 5$ Evaluate the expression by replacing $p$ with the first argument and $q$ with the second argument.

Order of Operations and Non-Commutativity

Standard addition and multiplication are commutative ($a + b = b + a$), but synthetic operators are rarely commutative:

• $a \star b \neq b \star a$ in most defined systems.
• When expressions include parentheses, such as $(2 \star 3) \star 4$, always evaluate the inner expression first:
  1. Inner term: $2 \star 3 = 3(2) - 2(3) + 5 = 6 - 6 + 5 = 5$.
  2. Outer term: $5 \star 4 = 3(5) - 2(4) + 5 = 15 - 8 + 5 = 12$.

Guess-and-Check vs. Deductive Elimination

Under examination conditions, unorganized guessing consumes valuable minutes. Replace blind trial-and-error with Systematic Bounded Testing:

1. Determine Boundaries: Identify the absolute minimum and maximum possible values.
2. Apply Parity and Divisibility: Rule out candidates that produce the wrong parity or fail divisibility constraints.
3. Record Intermediate Tests: Write down each tested value and whether the result was too high or too low to guide the next test logically.

Real-World Quantitative Reasoning in the Style of Official PEP Items

The Ministry describes the Ability Test as measuring a student's ability to "reason with words and quantities". Its own Ability Test practice items look less like puzzles and more like everyday decisions: choosing the better-value vendor and explaining why, deciding which player's scores make the team more likely to win, finding a total that can be shared equally among three colours, estimating time spent on a daily habit over a year, counting games in a competition where every team meets every other team once, and working out ages from family relationships. The skills below match those item types. The examples are original, so do not expect the same numbers on the test.

A Worked Decision Problem

Kerry-Ann has $300 to buy oranges for a class fair. Vendor A sells oranges at 5 for $100. Vendor B sells them at $18 each.

1. Vendor A: $300 buys $3$ groups of $5$, which is $15$ oranges, so each orange costs $20.
2. Vendor B: $300 divided by $18 is 16 remainder $12, so she can buy 16 oranges and has $12 change.
3. Vendor B gives more oranges for the same money because each orange is cheaper ($18 compared with $20).

When a question asks you to choose and explain, the explanation must use the numbers: "Vendor B, because each orange costs $18 instead of $20, so $300 buys 16 oranges instead of 15."