Can You Solve This? The Ultimate Order of Operations Math Puzzle

Visual math puzzles involving school supplies like pencils, rulers, and crayons are among the most viral brain teasers online. While they look simple at first glance, they test your ability to perform basic algebraic substitution and adhere strictly to the order of operations (PEMDAS/BODMAS).

In the puzzle image above, we need to determine the value of each school supply item across the top three equations to solve both the fourth (?) and fifth (??) equations.

Step-by-Step Breakdown

Let’s solve the system of equations line by line:

Line 1: Finding the Value of a Crayon

1 Crayon + 1 Crayon – 1 Crayon = 12
Simplifying: $\text{1 Crayon} = 12$
1 Crayon = 12

Line 2: Finding the Value of a Pencil

(1 Pencil $\times$ 1 Pencil) + 1 Pencil = 30
Let $p$ represent the value of a pencil:

$p^2 + p = 30$

$p^2 + p - 30 = 0$
Factoring the quadratic equation gives $(p + 6)(p - 5) = 0$ .
Since $p$ represents a positive count/value, $p = 5$ .
Verification: $(5 \times 5) + 5 = 25 + 5 = 30$ .
1 Pencil = 5

Line 3: Finding the Value of a Ruler

1 Ruler + (1 Ruler $\times$ 1 Ruler) = 20
Let $r$ represent the value of a ruler:

$r + r^2 = 20$

$r^2 + r - 20 = 0$
Factoring gives $(r + 5)(r - 4) = 0$ .
Taking the positive solution, $r = 4$ .
Verification: $4 + (4 \times 4) = 4 + 16 = 20$ .
1 Ruler = 4

Solving the Final Two Equations

Now that we have established the base values (Crayon = 12, Pencil = 5, Ruler = 4), let’s calculate both targets:

Line 4: Solving for “?”

\text{Pencil} \times \text{Ruler} - \text{Crayon} = ?

5 \times 4 - 12 = ?

Following the Order of Operations (PEMDAS/BODMAS):

Perform multiplication first: $5 \times 4 = 20$
Perform subtraction next: $20 - 12 = 8$

The value of “?” is 8!

Line 5: Solving for “??”

\text{Crayon} + \text{Pencil} \times \text{Ruler} = ??

12 + 5 \times 4 = ??