Add a draft about digit sum patterns.
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require 'victor'
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def sum_digits(n)
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while n > 9
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n = n.to_s.chars.map(&:to_i).sum
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end
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n
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end
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def step(x, y, n, dir)
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case dir
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when :top
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return [x,y+n,:right]
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when :right
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return [x+n,y,:bottom]
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when :bottom
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return [x,y-n,:left]
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when :left
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return [x-n,y,:top]
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end
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end
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def run_number(number)
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counter = 1
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x, y, dir = 0, 0, :top
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line_stack = [[0,0]]
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loop do
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x, y, dir = step(x,y, sum_digits(counter*number), dir)
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line_stack << [x,y]
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counter += 1
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break if x == 0 && y == 0
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end
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return make_svg(line_stack)
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end
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def make_svg(line_stack)
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line_length = 20
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xs = line_stack.map { |c| c[0] }
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ys = line_stack.map { |c| c[1] }
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x_offset = -xs.min
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y_offset = -ys.min
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svg_coords = ->(p) {
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nx, ny = p
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[(nx+x_offset)*line_length + line_length/2, (ny+y_offset)*line_length + line_length/2]
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}
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max_width = (xs.max - xs.min).abs * line_length + line_length
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max_height = (ys.max - ys.min).abs * line_length + line_length
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svg = Victor::SVG.new width: max_width, height: max_height
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style = { stroke: 'black', stroke_width: 5 }
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svg.build do
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line_stack.each_cons(2) do |pair|
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p1, p2 = pair
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x1, y1 = svg_coords.call(p1)
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x2, y2 = svg_coords.call(p2)
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line x1: x1, y1: y1, x2: x2, y2: y2, style: style
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circle cx: x2, cy: y2, r: line_length/6, style: style, fill: 'black'
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end
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end
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return svg
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end
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(1..9).each do |i|
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run_number(i).save "pattern_#{i}"
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end
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@ -0,0 +1,318 @@
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---
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title: Digit Sum Patterns and Modular Arithmetic
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date: 2021-12-30T15:42:40-08:00
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tags: ["Ruby", "Mathematics"]
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draft: true
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---
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When I was in elementary school, our class was briefly visited by our school's headmaster.
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He was there for a demonstration, probably intended to get us to practice our multiplication tables.
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_"Pick a number"_, he said, _"And I'll teach you how to draw a pattern from it."_
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The procedure was rather simple:
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1. Pick a number between 2 and 8 (inclusive).
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2. Start generating multiples of this number. If you picked 8,
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your multiples would be 8, 16, 24, and so on.
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3. If a multiple is more than one digit long, sum its digits. For instance, for 16, write 1+6=7.
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If the digits add up to a number that's still more than 1 digit long, add up the digits of _that_
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number (and so on).
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4. Start drawing on a grid. For each resulting number, draw that many squares in one direction,
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and then "turn". Using 8 as our example, we could draw 8 up, 7 to the right, 6 down, 5 to the left,
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and so on.
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5. As soon as you come back to where you started (_"And that will always happen"_, said my headmaster),
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you're done. You should have drawn a pretty pattern!
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Sticking with our example of 8, the pattern you'd end up with would be something like this:
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{{< figure src="pattern_8.svg" caption="Pattern generated by the number 8." class="tiny" alt="Pattern generated by the number 8." >}}
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Before we go any further, let's observe that it's not too hard to write code to do this.
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For instance, the "add digits" algorithm can be naively
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written by turning the number into a string (`17` becomes `"17"`), splitting that string into
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characters (`"17"` becomes `["1", "7"]`), turning each of these character back into numbers
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(the array becomes `[1, 7]`) and then computing the sum of the array, leaving `8`.
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{{< codelines "Ruby" "patterns/patterns.rb" 3 8 >}}
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We may now encode the "drawing" logic. At any point, there's a "direction" we're going - which
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I'll denote by the Ruby symbols `:top`, `:bottom`, `:left`, and `:right`. Each step, we take
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the current `x`/`y` coordinates (our position on the grid), and shift them by `n` in a particular
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direction `dir`. We also return the new direction alongside the new coordinates.
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{{< codelines "Ruby" "patterns/patterns.rb" 10 21 >}}
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The top-level algorithm is captured by the following code, which produces a list of
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coordinates in the order that you'd visit them.
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{{< codelines "Ruby" "patterns/patterns.rb" 23 35 >}}
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I will omit the code for generating SVGs from the body of the article -- you can always find the complete
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source code in this blog's Git repo (or by clicking the link in the code block above). Let's run the code on a few other numbers. Here's one for 4, for instance:
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{{< figure src="pattern_4.svg" caption="Pattern generated by the number 4." class="tiny" alt="Pattern generated by the number 4." >}}
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And one more for 2, which I don't find as pretty.
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{{< figure src="pattern_2.svg" caption="Pattern generated by the number 2." class="tiny" alt="Pattern generated by the number 2." >}}
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It really does always work out! Young me was amazed, though I would often run out of space on my
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grid paper to complete the pattern, or miscount the length of my lines partway in. It was only
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recently that I started thinking about _why_ it works, and I think I figured it out. Let's take a look!
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### Is a number divisible by 3?
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You might find the whole "add up the digits of a number" thing familiar, and for good reason:
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it's one way to check if a number is divisible by 3. The quick summary of this result is,
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> If the sum of the digits of a number is divisible by 3, then so is the whole number.
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For example, the sum of the digits of 72 is 9, which is divisible by 3; 72 itself is correspondingly
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also divisible by 3, since 24*3=72. On the other hand, the sum of the digits of 82 is 10, which
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is _not_ divisible by 3; 82 isn't divisible by 3 either (it's one more than 81, which _is_ divisible by 3).
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Why does _this_ work? Let's talk remainders.
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If a number doesn't cleanly divide another (we're sticking to integers here),
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what's left behind is the remainder. For instance, dividing 7 by 3 leaves us with a remainder 1.
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On the other hand, if the remainder is zero, then that means that our dividend is divisible by the
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divisor (what a mouthful). In mathematics, we typically use
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\\(a|b\\) to say \\(a\\) divides \\(b\\), or, as we have seen above, that the remainder of dividing
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\\(b\\) by \\(a\\) is zero.
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Working with remainders actually comes up pretty commonly in discrete math. A well-known
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example I'm aware of is the [RSA algorithm](https://en.wikipedia.org/wiki/RSA_(cryptosystem)),
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which works with remainders resulting from dividing by a product of two large prime numbers.
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But what's a good way to write, in numbers and symbols, the claim that "\\(a\\) divides \\(b\\)
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with remainder \\(r\\)"? Well, we know that dividing yields a quotient (possibly zero) and a remainder
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(also possibly zero). Let's call the quotient \\(k\\).
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{{< sidenote "right" "r-less-note" "Then, we know that when dividing \(b\) by \(a\) we have:" >}}
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It's important to point out that for the equation in question to represent division
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with quotient \(k\) and remainder \(r\), it must be that \(r\) is less than \(a\).
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Otherwise, you could write \(r = s + a\) for some \(s\), and end up with
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{{< latex >}}
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\begin{aligned}
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& b = ka + r \\
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\Rightarrow\ & b = ka + (s + a) \\
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\Rightarrow\ & b = (k+1)a + s
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\end{aligned}
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{{< /latex >}}
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In plain English, if \(r\) is bigger than \(a\) after you've divided, you haven't
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taken out "as much \(a\) from your dividend as you could", and the actual quotient is
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larger than \(k\).
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{{< /sidenote >}}
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{{< latex >}}
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\begin{aligned}
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& b = ka + r \\
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\Rightarrow\ & b-r = ka \\
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\end{aligned}
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{{< /latex >}}
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We only _really_ care about the remainder here, not the quotient, since it's the remainder
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that determines if something is divisible or not. From the form of the second equation, we can
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deduce that \\(b-r\\) is divisible by \\(a\\) (it's literally equal to \\(a\\) times \\(k\\),
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so it must be divisible). Thus, we can write:
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{{< latex >}}
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(b-r)|a
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{{< /latex >}}
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There's another notation for this type of statement, though. To say that the difference between
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two numbers is divisible by a third number, we write:
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{{< latex >}}
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b \equiv r\ (\text{mod}\ a)
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{{< /latex >}}
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Some things that _seem_ like they would work from this "equation-like" notation do, indeed, work.
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For instance, we can "add two equations":
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{{< latex >}}
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\textbf{if}\ a \equiv b\ (\text{mod}\ k)\ \textbf{and}\ c \equiv d, (\text{mod}\ k),\ \textbf{then}\
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a+c \equiv b+d\ (\text{mod}\ k).
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{{< /latex >}}
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Multiplying both sides by the same number (call it \\(n\\)) also works:
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{{< latex >}}
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\textbf{if}\ a \equiv b\ (\text{mod}\ k),\ \textbf{then}\ na \equiv nb\ (\text{mod}\ k).
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{{< /latex >}}
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Ok, that's a lot of notation and other _stuff_. Let's talk specifics. Of particular interest
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is the number 10, since our number system is _base ten_ (the value of a digit is multiplied by 10
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for every place it moves to the left). The remainder of 10 when dividing by 3 is 1. Thus,
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we have:
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{{< latex >}}
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10 \equiv 1\ (\text{mod}\ 3)
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{{< /latex >}}
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From this, we can deduce that multiplying by 10, when it comes to remainders from dividing by 3,
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is the same as multiplying by 1. We can clearly see this by multiplying both sides by \\(n\\).
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In our notation:
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{{< latex >}}
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10n \equiv n\ (\text{mod}\ 3)
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{{< /latex >}}
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But wait, there's more. Take any power of ten, be it a hundred, a thousand, or a million.
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Multiplying by that number is _also_ equivalent to multiplying by 1!
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{{< latex >}}
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10^kn = 10\times10\times...\times 10n \equiv n\ (\text{mod}\ 3)
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{{< /latex >}}
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We can put this to good use. Let's take a large number that's divisible by 3. This number
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will be made of multiple digits, like \\(d_2d_1d_0\\). Note that I do __not__ mean multiplication
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here, but specifically that each \\(d_i\\) is a number between 0 and 9 in a particular place
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in the number -- it's a digit. Now, we can write:
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{{< latex >}}
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\begin{aligned}
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0 &\equiv d_2d_1d_0 \\
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& = 100d_2 + 10d_1 + d_0 \\
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& \equiv d_2 + d_1 + d_0
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\end{aligned}
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{{< /latex >}}
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We have just found that \\(d_2+d_1+d_0 \\equiv 0\\ (\\text{mod}\ 3)\\), or that the sum of the digits
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is also divisible by 3. The logic we use works in the other direction, too: if the sum of the digits
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is divisible, then so is the actual number.
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There's only one property of the number 3 we used for this reasoning: that \\(10 \\equiv 1\\ (\\text{mod}\\ 3)\\). But it so happens that there's another number that has this property: 9. This means
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that to check if a number is divisible by _nine_, we can also check if the sum of the digits is
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divisible by 9. Try it on 18, 27, 81, and 198.
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Here's the main takeaway: __summing the digits in the way described by my headmaster is
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the same as figuring out the remainder of the number from dividing by 9__. Well, almost.
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The difference is the case of 9 itself: the __remainder__ here is 0, but we actually use 9
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to draw our line. We can actually try just using 0. Here's the updated `sum_digits` code:
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```Ruby
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def sum_digits(n)
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n % 9
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end
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```
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The results are similarly cool:
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{{< figure src="pattern_8_mod.svg" caption="Pattern generated by the number 8." class="tiny" alt="Pattern generated by the number 8 by just using remainders." >}}
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{{< figure src="pattern_4_mod.svg" caption="Pattern generated by the number 4." class="tiny" alt="Pattern generated by the number 4 by just using remainders." >}}
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{{< figure src="pattern_2_mod.svg" caption="Pattern generated by the number 2." class="tiny" alt="Pattern generated by the number 2 by just using remainders." >}}
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### Sequences of Remainders
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So now we know what the digit-summing algorithm is really doing. But that algorithm isn't all there
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is to it! We're repeatedly applying this algorithm over and over to multiples of another number. How
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does this work, and why does it always loop around? Why don't we ever spiral further and further
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from the center?
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First, let's take a closer look at our sequence of multiples. Suppose we're working with multiples
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of some number \\(n\\). Let's write \\(a_i\\) for the \\(i\\)th multiple. Then, we end up with:
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{{< latex >}}
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\begin{aligned}
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a_1 &= n \\
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a_2 &= 2n \\
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a_3 &= 3n \\
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a_4 &= 4n \\
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... \\
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a_i &= in
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\end{aligned}
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{{< /latex >}}
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This is actually called an [arithmetic sequence](https://mathworld.wolfram.com/ArithmeticProgression.html);
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for each multiple, the number increases by \\(n\\).
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Here's a first seemingly trivial point: at some time, the remainder of \\(a_i\\) will repeat.
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There are only so many remainders when dividing by nine: specifically, the only possible remainders
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are the numbers 0 through 8. We can invoke the [pigeonhole principle](https://en.wikipedia.org/wiki/Pigeonhole_principle) and say that after 9 multiples, we will have to have looped. Another way
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of seeing this is as follows:
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{{< latex >}}
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\begin{aligned}
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& 9 \equiv 0\ (\text{mod}\ 9) \\
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\Rightarrow\ & 9n \equiv 0\ (\text{mod}\ 9) \\
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\Rightarrow\ & 10n \equiv n\ (\text{mod}\ 9) \\
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\end{aligned}
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{{< /latex >}}
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The 10th multiple is equivalent to n, and will thus have the same remainder. The looping may
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happen earlier: the simplest case is if we pick 9 as our \\(n\\), in which case the remainder
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will always be 0.
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Repeating remainders alone do not guarantee that we will return to the center. The repeating sequence 1,2,3,4
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will certainly cause a spiral. The reason is that, if we start facing "up", we will always move up 1
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and down 3 after four steps, leaving us 2 steps below where we started. Next, the cycle will repeat,
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and since turning four times leaves us facing "up" again, we'll end up getting _further_ down.
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From this, we can devise a simple condition to prevent spiraling -- the _length_ of the sequence before
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it repeats _cannot be a multiple of 4_. This way, whenever the cycle restarts, it will do so in a
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different direction: backwards, turned once to the left, or turned once to the right. Clearly repeating
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the sequence backwards is guaranteed to take us back to the start. The same is true for the left and right-turn sequences,
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since after two iterations they will _also_ leave us facing backwards.
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Okay, so we want to avoid cycles with lengths divisible by four. What does it mean for a cycle to be of length _k_? It effectively means the following:
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{{< latex >}}
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\begin{aligned}
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& a_{k+1} \equiv a_1\ (\text{mod}\ 9) \\
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\Rightarrow\ & (k+1)n \equiv n\ (\text{mod}\ 9) \\
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\Rightarrow\ & kn \equiv 0\ (\text{mod}\ 9) \\
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\end{aligned}
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{{< /latex >}}
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If we could divide both sides by \\(k\\), we could go one more step:
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{{< latex >}}
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n \equiv 0\ (\text{mod}\ 9) \\
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{{< /latex >}}
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That is, \\(n\\) would be divisible by 9! This would contradict our choice of \\(n\\) to be
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between 2 and 8. What went wrong? Turns out, it's that last step: we can't always divide by \\(k\\).
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Some values of \\(k\\) are special, and it's only _those_ values that can serve as cycle lengths
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without causing a contradiction. So, what are they?
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They're values that have a common factor with 9. There are many numbers that have a common
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factor with 9; 3, 6, 9, 12, and so on. However, those can't all serve as cycle lengths: as we said,
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cycles can't get longer than 9. This leaves us with 3, 6, and 9 as _possible_ cycle lengths,
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none of which are divisible by 4. We've eliminated the possibility of spirals!
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{{< todo >}}
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This doesn't get to the bottom of it all.
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{{< /todo >}}
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### Generalizing to Arbitrary Divisors
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The trick was easily executable on paper because there's an easy way to compute the remainder of a number
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when dividing by 9 (adding up the digits). However, we have a computer, and we don't need to fall back on such
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cool-but-complicated techniques. To replicate our original behavior, we can just write:
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```
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def sum_digits(n)
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x = n % 9
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x == 0 ? 9 : x
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end
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```
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But now, we can change the `9` to something else. Any number we pick, so long as it isn't
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{{< sidenote "right" "div-4-note" "divisible by 4," >}}
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"Wait", you might be thinking, "I thought you said that 4 can't have a common factor with the divisor,
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and that means any even numbers are out, too."<br>
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<br>
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Good observation. Although the path-not-divisible-by-four condition is certainly sufficient, it is not
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<em>necessary</em>. There seems to be another, less restrictive, condition at play here: even numbers work fine. I haven't
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figured out what it is, but we might as well make use of it.
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{{< /sidenote >}} will work. I'll pick primes for good measure. Here are a few good ones from using 13
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(which corresponds to summing digits of base-14 numbers):
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{{< figure src="pattern_8_13.svg" caption="Pattern generated by the number 8 in base 14." class="tiny" alt="Pattern generated by the number 8 by summing digits." >}}
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{{< figure src="pattern_4_13.svg" caption="Pattern generated by the number 4 in base 14." class="tiny" alt="Pattern generated by the number 4 by summing digits." >}}
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Here are a few from dividing by 17 (base-18 numbers).
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{{< figure src="pattern_5_17.svg" caption="Pattern generated by the number 5 in base 18." class="tiny" alt="Pattern generated by the number 5 by summing digits." >}}
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Finally, base-30:
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{{< figure src="pattern_2_29.svg" caption="Pattern generated by the number 2 in base 30." class="tiny" alt="Pattern generated by the number 2 by summing digits." >}}
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{{< figure src="pattern_6_29.svg" caption="Pattern generated by the number 6 in base 30." class="tiny" alt="Pattern generated by the number 6 by summing digits." >}}
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|
|
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After Width: | Height: | Size: 5.9 KiB |
Loading…
Reference in New Issue