Pythagorean triples are sets of three integers that are solutions to the associated identity for right triangles.
The question is this:
Can you find a formula or algorithm for generating Pythagorean Triples?
There may be more than one formula, and each might not be exhaustive. All we are after is a set of rules such that, given one number, can you find the other two that belong to the triplet?
There is an island with 100 women. 50 of the women have red dots on their foreheads, and the other 50 women have blue dots on their foreheads.
If a woman ever learns the color of the dot on her forehead, she must permanently leave the island in the middle of that night.
One day, an oracle appears and says “at least one woman has a blue dot on her forehead.” The woman all know that the oracle speaks the truth.
All the woman are perfect logicians (and know that the others are pefect logicians too). What happens next?
Three Masters of Logic wanted to find out who was the wisest amongst them. So they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old sage said. “I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the color of the dot on his forehead first, wins.” And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: “I have a red dot on my forehead.”
How did he guess?
Step 1: Use long division (no calculators please!) to convert the fractions (in this case, specifically reciprocal integers) to decimal representations.
Step 2: Look at the results. What do you notice? Write down all the questions that come up.
Step 3: As a group, we will make a list of all the questions.
Step 4: Think and write up your answers to the questions.
Step 5: Discuss!
Note: Work with a partner to divide up the work in Stages 1, 2, and 4.
1/2 = __________ 1/11 = __________
1/3 = __________ 1/12 = __________
1/4 = __________ 1/13 = __________
1/5 = __________ 1/14 = __________
1/6 = __________ 1/15 = __________
1/7 = __________ 1/16 = __________
1/8 = __________ 1/17 = __________
1/9 = __________ 1/18 = __________
1/10 = __________ 1/19 = __________
1/20 = __________