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Ch 06 - Algebra Play8th Ganita Prakash Part 2

Basic 'Think of a Number' Algebraic Translation

PROCEDURAL Formula

\frac{2x + 4}{2} - x = 2

State the final algebraic equation when you think of a number x, double it, add 4, divide by 2, and subtract the original number.

Derivation

Algorithmic SOP:

Let the starting number be $x$ .
Double it: $2x$ .
Add 4: $2x + 4$ .
Divide the entire binomial by 2: $\frac{2x + 4}{2} = x + 2$ .
Subtract the original number: $x + 2 - x = 2$ .

Component Anatomy:

$x$ : The initial algebraically unknown number chosen.
$2$ : The constant final answer, strictly independent of $x$ .

Boundary Conditions / Domain:

$x$ can be absolutely any real number ( $x \in \mathbb{R}$ ). The algebraic cancellation guarantees the result will always be 2.

Prism Memory Hook

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Date Guessing Trick (Algebraic Expression)

PROCEDURAL Formula

100M + 165 + D = R

Write the master algebraic equation representing the 'Date Guessing Trick' after all operations are applied to the Month (M) and Day (D).

Derivation

Algorithmic SOP:

Multiply month by 5: $5M$ .
Add 6: $5M + 6$ .
Multiply by 4: $4(5M + 6) = 20M + 24$ .
Add 9: $20M + 33$ .
Multiply by 5: $5(20M + 33) = 100M + 165$ .
Add the day: $100M + 165 + D = R$ .

Component Anatomy:

$M$ : The numerical value of the Month.
$D$ : The numerical value of the Day.
$R$ : The final numerical result given by the user.

Boundary Conditions / Domain:

$M \in \{1, 2, \dots, 12\}$ strictly.
$D \in \{1, 2, \dots, 31\}$ strictly, bound by the respective calendar month.

Prism Memory Hook

100 Months plus a Day

Decoding the Date Guessing Trick

How do you algebraically extract the exact Month (M) and Day (D) from the final result (R) in the Date Guessing trick?

PROCEDURAL

Step 1: Subtract the constant $165$ from the user's Final Result ( $R - 165$ ).
Step 2: The resulting number is structurally equivalent to $100M + D$ .
Step 3: The last two digits strictly represent the Day ( $D$ ), and the remaining digit(s) preceding them strictly represent the Month ( $M$ ).

Boundary Conditions / Domain:

$R$ must be a valid outcome of the algorithm, meaning $R \ge (100(1) + 165 + 1) = 266$ .

Prism Memory Hook

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Also in this chapter

28 more cards to study in Prism

42-Digit Reversal Difference (Algebraic Form)
5Divisibility of 2-Digit Reversal Difference
62-Digit Reversal Sum (Algebraic Form)
7Divisibility of 2-Digit Reversal Sum
83-Digit Cyclic Sum Expansion
9Divisibility of 3-Digit Cyclic Sums
10The 6-Digit Repeated Sequence Trick
11The Largest Product Algorithm (3 Digits)
12Algebraic Proof of the Largest Product
13Sum Sequence of Odd Numbers Fractions
14Number Pyramid Base Rule
15Topmost Value of a 3-Row Number Pyramid
16Topmost Value of a 4-Row Number Pyramid
17Virahānka-Fibonacci Sequence
18Virahānka-Fibonacci Pyramids
19Calendar Magic: 2x2 Grid Sum
20Decoding the 2x2 Calendar Grid
21Calendar Magic: 3x3 Grid Sum
223x3 Calendar Grid: Center Element Corollary
23Calendar Magic: 5-Cell Cross Grid Sum
24Heads and Legs Algebraic Setup
25Heads and Legs Arithmetic Shortcut
26Simultaneous Linear Equations: Cowherd Transfer (Setup)
27Linear Profit-Volume Function
28Magical Ponds Iterative Setup
29Karim and the Genie: Iterative Expansion
30Genie Coin Doubling: Wealth Increase Inequality
31Genie Coin Doubling: Absolute Bankruptcy Condition

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8th Ganita Prakash Part 2

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