Mechanism 03
Algebraic Cancellation
The starting number is not guessed. It is carried through the procedure, then removed by the final instruction.
Three small pieces of language
Meet them in plain language before they appear in the proof.
x- A temporary name for whatever whole number you chose. We can follow its role without knowing its value.
- Invariant
- Something that stays the same while other parts of a system change. Here, the final result stays 5.
- Cancellation
- Equal amounts with opposite signs combine to zero. Near the end,
x − xdisappears for that reason.
Follow the role, not the secret
Each expression is the same journey viewed after one more instruction.
x2x2x + 10x + 55
xYour original number receives a name.2xDoubling makes two copies of that number.2x + 10Ten joins the doubled number.(2x + 10) / 2 = x + 5Halving affects both parts: two copies become one, and ten becomes five.x + 5 − x = 5Subtracting the original createsx − x. Those equal opposites cancel.
The private value moves. The destination does not.
Amber follows the changing state; blueprint blue marks inspectable structure.