How to Count like a Mathematician Counting Warmup Round 1 squares - PowerPoint PPT Presentation

How to Count like a Mathe- matician Lara Pudwell Introduction How to Count like a Mathematician Counting Warmup Round 1 squares Lara Pudwell dominoes Round 2 trees Department of Mathematics and Computer Science blocks The Bigger Picture VERUM Seminar So what do June 7, 2013 you research?

What do you research? How to Count like a Mathe- matician Lara Pudwell Introduction Counting Warmup enumerative combinatorics Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

What do you research? How to Count like a Mathe- matician Lara Pudwell Introduction Counting Warmup enumerative combinatorics Round 1 squares dominoes Round 2 counting things that are hard to count trees blocks The Bigger Picture So what do you research?

So you think you can count? How to Count like a Mathe- matician Lara Pudwell Introduction 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . . Counting Warmup Round 1 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , . . . squares dominoes Round 2 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 , 50 , . . . trees blocks The Bigger Picture So what do you research?

So you think you can count? How to Count like a Mathe- matician Lara Pudwell Introduction 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . . Counting Warmup Round 1 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , . . . squares dominoes Round 2 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 , 50 , . . . trees blocks How many? The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count like a Mathe- matician How many dots do you see? Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count like a Mathe- matician How many dots do you see? Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees 6 + 5 + 6 blocks The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count like a Mathe- matician How many dots do you see? Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees 6 + 5 + 6 = 17 blocks The Bigger Picture So what do you research? Trick 1: Break your counting job into smaller jobs and add those results.

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture 7 · 6 So what do you research?

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture 7 · 6 = 42 So what do you research? Trick 2: Multiplying speeds things up!

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks 7 · 6 The Bigger Picture So what do you research?

You’ve already got some tricks up your sleeve.... How to Count How many dots do you see? like a Mathe- matician Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks 7 · 6 3 − The Bigger Picture 42 − 3 = 39 So what do you research? Trick 3: Count a larger set and subtract off the extras.

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 What’s the pattern? trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 What’s the pattern? trees blocks Count (end in ) + (end in ). The Bigger Picture So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 What’s the pattern? trees blocks Count (end in ) + (end in ). The Bigger Picture (end in ) = (end in ) = (lines with one less square). So what do you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 What’s the pattern? trees blocks Count (end in ) + (end in ). The Bigger Picture (end in ) = (end in ) = (lines with one less square). So what do Lines of length a = 2 × (lines of length a − 1) you research?

Thinking in black and white How to Count You have black squares and white squares . like a Mathe- matician How many ways can you arrange a line of squares? Lara Pudwell 1 square? 2 ways Introduction 2 squares? 4 ways Counting Warmup 3 squares? 8 ways Round 1 squares dominoes Round 2 What’s the pattern? trees blocks Count (end in ) + (end in ). The Bigger Picture (end in ) = (end in ) = (lines with one less square). So what do Lines of length a = 2 × (lines of length a − 1) you research? length 1 2 3 4 5 6 7 8 # of arrangements 2 4 8 16 32 64 128 256

Thinking in black and white How to Count What if you can’t have two black squares in a row? like a Mathe- matician 1 square? 2 ways Lara Pudwell Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count What if you can’t have two black squares in a row? like a Mathe- matician 1 square? 2 ways Lara Pudwell 2 squares? 3 ways Introduction Counting Warmup Round 1 squares dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count What if you can’t have two black squares in a row? like a Mathe- matician 1 square? 2 ways Lara Pudwell 2 squares? 3 ways Introduction 3 squares? 5 ways Counting Warmup Round 1 squares What’s the pattern? dominoes Round 2 trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count What if you can’t have two black squares in a row? like a Mathe- matician 1 square? 2 ways Lara Pudwell 2 squares? 3 ways Introduction 3 squares? 5 ways Counting Warmup Round 1 squares What’s the pattern? dominoes Round 2 Count (end in ) + (end in ). trees blocks The Bigger Picture So what do you research?

Thinking in black and white How to Count What if you can’t have two black squares in a row? like a Mathe- matician 1 square? 2 ways Lara Pudwell 2 squares? 3 ways Introduction 3 squares? 5 ways Counting Warmup Round 1 squares What’s the pattern? dominoes Round 2 Count (end in ) + (end in ). trees (end in ) = (end in ) = (lines with two less squares) blocks The Bigger (end in ) = (lines with one less square). Picture So what do you research?