Formula Detetctive

Transcript

Formula DetectiveThink of each number machine as a robot with a secret rule. It takes the two numbers at the top, does something with them based on its secret rule, and gives you the result at the bottom. Our job is to be the detective and figure out that one secret rule that works for every example. How to be a Formula Detective Look at the first example: Take the first set of inputs (top numbers) and the output (bottom number). Guess a simple rule: Start with simple math. Did the machine add them? Subtract them? Multiply them? Test your rule: Use your guessed rule on the next example. If it works, you are probably right! Test it on one more to be sure. If it fails, your guess was wrong. Go back to Step 2 and try a new rule. Let's Solve the Puzzles First Machine (Green) Example 1: Inputs are 5 and 2. Output is 8. Guess: Let's try adding: 5 + 2 = 7. Nope. Let's try multiplying the first number by 2: 2 × 5 = 10. That's close! To get from 10 to 8, we subtract 2. Hey, 2 is the second number! Our Rule Guess: "Double the first number, then subtract the second number." Formula: 2a – b Test on Example 2: Inputs are 8 and 1. Output is 15. Let's use our rule: 2 × 8 - 1 = 16 - 1 = 15. It works! Conclusion: The formula is correct. Expressions and Final Answer: Expression for (5, 2): 5 + 2 - 2 = 5 Expression for (8, 1): 8 + 1 - 2 = 7 Expression for (9, 11): 9 + 11 - 2 = 18 Expression for (11, 10): 11 + 10 - 2 = 19 (Note: the image output of 18 seems to be a typo, as 19 follows the rule). The formula is: a + b - 2 The expression for (a, b) gives an output of: a + b - 2 Second Machine (Blue) Example 1: Inputs are 5 and 2. Output is 5. Guess: Adding them gives 5 + 2 = 7. To get to 5, we need to subtract 2. Our Rule Guess: "Add the two numbers, then subtract 2." Formula: a + b – 2 Test on Example 2: Inputs are 8 and 1. Output is 7. Let's use our rule: 8 + 1 - 2 = 9 - 2 = 7. It works! Test on Example 3: Inputs are 9 and 11. Output is 18. Let's use our rule: 9 + 11 - 2 = 20 - 2 = 18. It works perfectly! Third Machine (Purple) Example 1: Inputs are 4 and 1. Output is 5. Guess 1: Adding them gives 4 + 1 = 5. Looks good so far! Test on Example 2: Inputs are 6 and 0. Output is 1. Let's use our rule a + b: 6 + 0 = 6. This is not 1. So our guess was wrong. Guess 2: Let's go back to the first example (4, 1 -> 5). What if we multiply the numbers? 4 × 1 = 4. To get to 5, we need to add 1. Our New Rule Guess: "Multiply the two numbers, then add 1." . Formula: a × b + 1 or ab + 1 Test on Example 2 again: Inputs are 6 and 0. Output is 1. Let's use our new rule: 6 × 0 + 1 = 0 + 1 = 1. It works! Test on Example 3: Inputs are 3 and 2. Output is 7. Let's use our rule: 3 × 2 + 1 = 6 + 1 = 7. It works! Expressions and Final Answer: Expression for (4, 1): 4 × 1 + 1 = 5 Expression for (6, 0): 6 × 0 + 1 = 1 Expression for (3, 2): 3 × 2 + 1 = 7 The formula is: a × b + 1 For inputs (10, 3) the output would be: 10 × 3 + 1 = 31 The expression for (a, b) gives an output of: ab + 1