Algebraic Formulas and Patterns

Writing Formulas

Rules as Equations

A formula is a general rule expressed as an algebraic expression, establishing constant relationships between different quantities.
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Key Points

  1. Formulas use variables to state general mathematical rules compactly.
  2. Example: The perimeter of a square is written as the formula P=4sP = 4s.
  3. In this formula, PP represents the perimeter and ss is the side length.
  4. Using formulas lets us solve real-world problems by plugging in values.

Olympiad Fact

A formula is a specific type of algebraic identity; it sets up a constant, provable relationship that is highly useful for multi-step problem solving.

Patterns through Numbers

Finding the General Rule

We can write a general algebraic rule using a variable nn to represent any term in a sequence based on its position or step number.
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Key Points

  1. Algebra can describe repeating number sequences in a general way.
  2. We use the variable nn to represent the position or step number.
  3. Example: The sequence 3,6,9,12,…3, 6, 9, 12, \dots increases by 33 each time.
  4. The general term for the nthn^{\text{th}} position in this pattern is 3n3n.

Olympiad Fact

To find which step will have a value of 7575 in a 2n+32n+3 pattern, set up the equation 2n+3=752n+3 = 75, yielding 2n=722n = 72, which simplifies to step n=36n = 36!

Number Machines

Input-Output Rules

A number machine takes inputs, applies a consistent algebraic rule, and outputs a calculated value. This rule is represented as an algebraic expression.
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Key Points

  1. Number machines take input variables and apply a set algebraic rule.
  2. They produce a unique, calculated output for each set of inputs.
  3. The inputs are represented by variables like xx and yy.
  4. Example: The rule 'Two times the first number + the second' is written as 2x+y2x+y.

Olympiad Fact

In Olympiad pattern tests, always test your algebraic rule against multiple input-output pairs to verify that the pattern holds true in all cases.

Calendar 2×22 \times 2 Square Pattern

Algebraic Calendar Magic

A 2×22 \times 2 grid of dates in any calendar has a fixed algebraic structure based on the starting date aa, allowing us to predict sum totals.
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Key Points

  1. A 2×22 \times 2 calendar grid always follows a constant algebraic structure.
  2. If aa is the top-left date, the four dates are aa, a+1a+1, a+7a+7, and a+8a+8.
  3. The sum of all four numbers in the grid is always equal to 4a+164a + 16.
  4. The sum of the diagonal numbers is always equal to 2a+82a + 8.

Olympiad Fact

To find the sum of any 2×22 \times 2 calendar grid, simply take the top-left number aa, multiply it by 44, and add 1616—a classic mental math shortcut!

Calendar Plus-Shape Pattern

Symmetrical Centered Sums

A plus-sign shaped group of 5 dates on a calendar has a highly symmetrical structure. The sum of these 5 dates is always exactly 5 times the center date.
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Key Points

  1. A plus-shape calendar pattern selects five symmetrically arranged dates.
  2. If aa is the central date, the surrounding dates are a−7a-7, a−1a-1, a+1a+1, and a+7a+7.
  3. Adding all five dates simplifies algebraically: (a−7)+(a−1)+a+(a+1)+(a+7)(a-7) + (a-1) + a + (a+1) + (a+7).
  4. The sum of the five dates is always exactly 55 times the center number.

Olympiad Fact

This cancellation pattern is a beautiful real-world example of symmetric algebraic balance, commonly tested in competitive exams to teach grouping.

Matchstick Patterns

Linear Geometric Growth

We can represent the growth of repeating matchstick shapes using a linear algebraic rule, allowing us to find the sticks needed for any step.
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Key Points

  1. Matchstick patterns show how geometric shapes grow step-by-step.
  2. We use step number nn to write a general rule for the sticks needed.
  3. Example: A square matchstick pattern grows as 4,7,10,13,…4, 7, 10, 13, \dots.
  4. Each new square adds 33 sticks, represented by the linear algebraic rule 3n+13n + 1.

Olympiad Fact

For any matchstick sequence with starting value SS and constant difference dd, the general algebraic rule is written in the form dn+(S−d)dn + (S - d).

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