9.11 Summary & Closing Puzzle
Summary
Key Points
- The square on the diagonal of a square holds double the area; this is how Baudhayana doubled a square.
- In a right triangle, the side opposite the right angle is the hypotenuse, and it is always the longest side.
- For an isosceles right triangle with equal sides a and hypotenuse c: c^2 = 2a^2, so c = a\sqrt2.
- The number \sqrt2 lies between 1.414 and 1.415. It cannot be a terminating decimal, and it cannot be a fraction \tfrac{m}{n}, it is irrational.
- Baudhayana–Pythagoras Theorem: for any right triangle with legs a, b and hypotenuse c, a^2 + b^2 = c^2.
- A trio of positive integers with a^2 + b^2 = c^2 is a Baudhayana (Pythagorean) triple, e.g. (3,4,5), (5,12,13), (8,15,17). If (a,b,c) is a triple so is (ka,kb,kc), giving infinitely many.
- The formula a = m^2-n^2,\ b = 2mn,\ c = m^2+n^2 generates every triple; a triple with no common factor >1 is primitive, and every triple is a scaled primitive.
- The distance between grid points is \sqrt{(\text{horizontal step})^2 + (\text{vertical step})^2}.
- Fermat’s Last Theorem: a^n + b^n = c^n has no positive-integer solution for n > 2. Proved by Andrew Wiles in 1994.
A Closing Puzzle: The Mixed-Up Coin Bags
Try This A merchant has three sealed cloth bags. One holds only gold coins, one only silver, one a mix of gold and silver. The bags wear tags reading GOLD, SILVER, MIXED, but the merchant warns that not a single tag is correct. You are allowed to reach into just one bag and pull out a single coin without peeking inside. Can you correctly re-tag all three? (Hint: draw from the bag tagged MIXED. Since its tag is wrong, that bag is pure, so the one coin you draw fixes its true contents; then the “all tags wrong” rule pins down the remaining two.)