LS POW �Iterations

May 2024

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Thinking this week from:

Vinny, Sean, Oskar, Cole, Nora, Lucia, Noah, Sophia, Yolanda, Mukang, Ada

What is the LS Problem Solving Club?

• Once a week (or so!) we'll offer a time for students who love puzzling and problem-solving to have the chance to come together to do just that, puzzle and solve!
• The only prerequisite is that you work on the preview puzzle in advance. You do not need to solve it necessarily but you should have a good start. You can work on your own or with a grown-up, but you must have your thinking recorded (neatly!) on a separate piece of paper and you must be willing/able to explain your reasoning to others.
• We'll come together to talk about solving strategies and then tackle more challenging problems.
• Students who complete the weekly puzzle will have their work shared on a slide show that can be viewed from home.

Problem 1

Kaprekar's Constant

Kaprekar’s Routine

Iteration means doing something over and over. It's a very important idea in math and computer science. In 1949 the mathematician D. R. Kaprekar from India came up with an iteration that features a certain "magic number." 

Here's how it works:

1. Pick any four-digit number, as long as the digits are not all the same. (Don't use 2,222, etc.) 
2. Rearrange the digits to make the largest possible number with those digits, and the smallest possible (positive, whole) number.
3. Then subtract the smallest from the largest.
4. And then iterate! Do it all over again with that number!

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Preview Puzzle

Pick a starting four-digit number and use Kaprekar's Routine to iterate at least eight times. What do you notice?

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Try again with another four digit number. Think about what might be happening and why. Keep a neat record of your work. Be ready to share your thinking!

Article from The Guardian: 6174 (Kaprekar's Constant)

Problem 1

1. What is Kaprekar's magic number? Will every four-digit number lead to the same magic number? Why do you think this happens?

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• What will happen if you try this game with three- digit numbers? With five-digit numbers?

Problem 2

Squaring the Digits

Squaring the Digits

Start with any three- or four-digit number. Square each digit (multiply it by itself), and then add up the squares. Repeat! 

Look at the example begun here:

347 -> 9 + 16 + 49 = 74 

74 -> 49 + 16 = 65

65 -> 36 + 25 = 61

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Problem 2

1. What eventually happens as you do this over and over?
2. What will happen if you start with a different number? 
3. What if you start with a number with less than three digits or more than four?

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Can you explain what's going on?!