The missing two cake strips
Pip bought a bigger square cake. Two whole strips have vanished from his invoice.
The formula in this story
Story 4 of 8
What you will take away
Find both ab strips and explain where 2ab comes from.
Read one piece at a time. There is no timer. “Help” stays beside you throughout.
Before reading, how does it feel? (optional)
See the five reading steps
Choose a section directly
What do the words mean?
A square a² means a × a. The area of a rectangle is width times height. Expansion rewrites a product as a sum.
What are real numbers?
Real numbers are the numbers on the number line: integers, fractions and numbers such as √2. Our formulas apply to these numbers, even when the cake picture uses only positive lengths.
Want a hand with the prerequisites?The bakery with missing buns
A suspicious plan
Pip: “Side a plus b, so the cake must be a² plus b².” Mika discovers two long crumbs on his beak.
A square means a side times itself: (a + b)(a + b).
The clue
A cut at a in each direction makes four pieces: a square, a strip, another strip, and a small square.
Cut at a in both directions. The four areas are a², ab, ba and b².
The twist
Both strips have area ab. They were never optional extras. Pip returns them, very reluctantly.
The two strips have equal area: ab + ba = 2ab. Nothing was added to the cake.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
By the definition of a square, (a + b)² = (a + b)(a + b).
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For all real a, b, (a + b)² = a² + 2ab + b². The area picture uses nonnegative lengths.
- By the definition of a square, (a + b)² = (a + b)(a + b).
- Distribute twice: a(a + b) + b(a + b) = a² + ab + ba + b².
- Real multiplication commutes, so ba = ab. Combining the two strips gives 2ab. This algebra covers negative values too.
Pip has one more idea…
For a = 3 and b = 2, how much area did a² + b² forget?
Read the explanation for 6: one strip
There are two 3 × 2 strips. One strip is 6; together they are 12.
Read the explanation for 12: two strips
Yes: 25 − (9 + 4) = 12 = 2 × 3 × 2. The missing pieces are back.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
How does it feel now? (optional)
Your record for this story
Your first answers and your latest answers will appear here.
Where does your curiosity go next?
No score, no locked doors. You can return to any definition.
These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.