{"id":263,"date":"2024-08-22T19:18:11","date_gmt":"2024-08-22T19:18:11","guid":{"rendered":"https:\/\/learnx.ca\/mt\/?page_id=263"},"modified":"2025-11-30T15:12:22","modified_gmt":"2025-11-30T15:12:22","slug":"infinity-coding","status":"publish","type":"page","link":"https:\/\/learnx.ca\/mt\/infinity-coding\/","title":{"rendered":"CODING PUZZLES"},"content":{"rendered":"\n

[Solutions<\/a> are shown after the puzzles]<\/p>\n\n\n\n

Puzzle 1: Infinity (Scratch)<\/h4>\n\n\n\n

Go to https:\/\/scratch.mit.edu\/projects\/964712868\/editor<\/a> .<\/p>\n\n\n\n

Run the code to see the list of numbers shown below.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

1. <\/strong>The code finds the sum of 10 fractions.<\/p>\n\n\n\n

a) <\/strong>Which fractions are they?<\/span><\/p>\n\n\n\n

b)<\/strong> What is the sum if this pattern continues forever?<\/p>\n\n\n\n

2.<\/strong> How does the code work?<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

3.<\/strong> Edit the code as shown on the right to find the sum of fractions 1\/4, 1\/16, 1\/64, and so on. <\/p>\n\n\n\n

a)<\/strong> What is the sum?<\/span><\/p>\n\n\n\n

b) <\/strong>How does this sum make sense?<\/p>\n\n\n\n

4. <\/strong>Edit the code to find the sum of fractions 1\/2, 1\/8, 1\/32, and so on.<\/p>\n\n\n\n

a)<\/strong> What is the sum?<\/span><\/p>\n\n\n\n

b) <\/strong>How does this sum make sense?<\/p>\n\n\n\n

\n\n\n\n

Puzzle 2: Infinity (Python)<\/h4>\n\n\n\n

Go to https:\/\/cscircles.cemc.uwaterloo.ca\/console<\/a> .<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

Enter and run this Python code to list the numbers shown on the right.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

1. <\/strong>The code finds the sum of 10 fractions.<\/p>\n\n\n\n

a) <\/strong>Which fractions are they?<\/span><\/p>\n\n\n\n

b)<\/strong> What is the sum if this pattern continues forever?<\/p>\n\n\n\n

2.<\/strong> How does the code work?<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

3.<\/strong> Edit the code as shown on the right to find the sum of fractions 1\/4, 1\/16, 1\/64, and so on. <\/p>\n\n\n\n

a)<\/strong> What is the sum?<\/span><\/p>\n\n\n\n

b) <\/strong>How does this sum make sense?<\/p>\n\n\n\n

4. <\/strong>Edit the code to find the sum of fractions 1\/2, 1\/8, 1\/32, and so on.<\/p>\n\n\n\n

a)<\/strong> What is the sum?<\/span><\/p>\n\n\n\n

b) <\/strong>How does this sum make sense?<\/p>\n\n\n\n

\n\n\n\n

Puzzle 3: Natural Density<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8 \u2026}<\/p>\n\n\n\n

Even numbers = {2, 4, 6, 8, 10 \u2026}<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

Natural density<\/em> <\/strong>of even numbers <\/p>\n\n\n\n

= Chance of picking an even number from the infinite set of natural numbers<\/p>\n\n\n\n

= 0.5<\/p>\n\n\n\n

Puzzles<\/strong><\/p>\n\n\n\n

    \n
  1. What is the natural density of the odd numbers: {1, 3, 5, 7, 9 …}?<\/li>\n\n\n\n
  2. What is the natural density of multiples of 3: {3, 6, 9, 12, 15 …}?<\/li>\n\n\n\n
  3. What is the natural density of multiples of 10: {10, 20, 30, 40, 50 …}?<\/li>\n\n\n\n
  4. What is the natural density of the number 1?<\/li>\n\n\n\n
  5. What is the natural density of the square numbers: {1, 4, 9, 16, 25 …}?<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    SOLUTIONS<\/h1>\n\n\n\n

    Puzzle 1<\/h4>\n\n\n\n

    1.a)<\/strong>  1\/2, 1\/4, 1\/8, 1\/16 \u2026     <\/p>\n\n\n\n

    1.b)<\/strong>  1  [we see in the image below that all these fractions fit in 1 square]<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    3.<\/strong>  1\/3          4.<\/strong> 2\/3<\/p>\n\n\n

    \n
    \"\"<\/figure>\n<\/div>\n\n\n

    In the 2 images on the right, we see that:<\/p>\n\n\n\n