Snakes!

src/Warm-Ups/Snakes/main.typ

@@ -0,0 +1,89 @@
+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.4.2"
+
+#show: handout.with(
+  title: [Warm-Up: Snakes!],
+  by: "Mark",
+  page_numbers: false,
+)
+
+#box(place(box(width: 100%, height: 8in, align(horizon, image(
+  "7.svg",
+  width: 360mm,
+)))))
+
+#problem()
+Evaluate this integral. \
+All integrals are of the form $integral_a^b 1 #h(1mm) d x$.
+
+#if_solutions(pagebreak())
+
+#solution[
+  This integral is drawn recursively with the following code: \
+
+  #v(2mm)
+
+  #let snake(depth, x, y) = {
+    if depth == 0 {
+      math.attach(math.integral, br: x, tr: y)
+    } else {
+      let bot = snake(depth - 1, x, "1")
+      let top = snake(depth - 1, "0", y)
+      snake(depth - 1, bot, top)
+    }
+  }
+
+  ```typst
+  #let snake(depth, x, y) = {
+    if depth == 0 {
+      $integral_#x ^#y$
+    } else {
+      let bot = snake(depth - 1, x, "1")
+      let top = snake(depth - 1, "0", y)
+      snake(depth - 1, bot, top)
+    }
+  }
+  ```
+
+  #v(5mm)
+
+  For example:
+
+  $ snake\(0, x, y) = #snake(0, "x", "y") $
+
+  #v(2mm)
+
+  $ snake\(1, x, y) = #snake(1, "x", "y") $
+
+  #v(2mm)
+
+  $ snake\(2, x, y) = #snake(2, "x", "y") $
+
+  In other words, we want $f_7 (0, 1)$, where...
+  - $f_0(x, y) := integral_x^y$
+  - $f_(n+1)(x, y) := f_n [f_n (x, 1), f_n (0, y)]$
+
+  #v(5mm)
+  Expanding a few iterations, we find:
+  - $f_0(x, y) = integral_x^y = y-x$
+  - $f_1(x, y) = f_0 [f_0 (x, 1), f_0 (0, y)] = y+x-1$
+  - $f_2(x, y) = f_1 [f_1 (x, 1), f_1 (0, y)] = y+x-2$
+
+  #v(5mm)
+
+  We can use induction to show that this pattern continues. \
+  If $g_n = y+x+c$, then $g_(n+1) = y+x+(3c+1)$.
+
+
+  #v(5mm)
+  Finally, use this recursion to find that
+  $f_0, f_1, ..., f_7 = 1, 0, -1, -4, -13, -40, -121, -364$
+
+  One can also find an explicit formula for $g_n$:
+
+  $
+    f_(n+1) = g_n & = x + y + 3^n c + 3^0 + 3^1 + ... + 3^n \
+                  & = x + y + 3^n c + sum_(i=0)^n 3^i \
+                  & = x + y + 3^n c + (3^(n+1) + 1)/2
+  $
+]