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Day 17 part 1-2
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‎day17/input_test‎

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target area: x=20..30, y=-10..-5

‎day17/part1.zig‎

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const std = @import("std");
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const sqrt = std.math.sqrt;
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const abs = std.math.absInt;
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const max = std.math.max;
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const min = std.math.min;
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fn floor(x: f64) i32 {
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return @floatToInt(i32, @floor(x));
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}
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fn ceil(x: f64) i32 {
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return @floatToInt(i32, @ceil(x));
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}
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fn float(x: i32) f64 {
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return @intToFloat(f64, x);
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}
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fn xExists(y: i32, s: i32, x1: i32, x2: i32) bool {
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// Solve for x:
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// if s < x:
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// x1 <= sum x-s+1 .. x <= x2 <==>
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// x1 <= s*x - (s-1)*s/2 <= x2 <==>
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// x1/s + (s - 1)/2 <= x <= x2/s + (s - 1)/2
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{
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const min_sx = x1 + @divExact((s - 1) * s, 2);
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const max_sx = x2 + @divExact((s - 1) * s, 2);
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if (min_sx <= max_sx and s * s < min_sx) {
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return true;
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}
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}
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// if s <= x:
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// x1 <= (x*x + x) / 2 <= x2
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// Solve:
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// sx1 = (-1 - sqrt(1 + 8*x1)) / 2
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// sx2 = (-1 + sqrt(1 + 8*x1)) / 2
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//
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// sx3 = (-1 - sqrt(1 + 8*x2)) / 2
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// sx4 = (-1 + sqrt(1 + 8*x2)) / 2
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//
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// so:
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// (x <= sx1 or sx2 <= x) and
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// (sx3 <= x <= sx4) <==>
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// (max(sx3, s) <= x <= min(sx1, sx4)) or
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// (max(sx2, sx3, s) <= x <= sx4)
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{
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const sx1 = floor((-1 - sqrt(float(1 + 8 * x1))) / 2);
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const sx2 = ceil((-1 + sqrt(float(1 + 8 * x1))) / 2);
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const sx3 = ceil((-1 - sqrt(float(1 + 8 * x2))) / 2);
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const sx4 = floor((-1 + sqrt(float(1 + 8 * x2))) / 2);
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{
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const min_x = max(sx3, 0);
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const max_x = min(min(sx1, sx4), s);
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if (min_x <= max_x) {
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return true;
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}
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}
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{
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const min_x = max(max(sx2, sx3), 0);
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const max_x = min(sx4, s);
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if (min_x <= max_x) {
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return true;
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}
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}
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}
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return false;
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}
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pub fn main() !void {
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var file = try std.fs.cwd().openFile("input", .{});
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defer file.close();
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var buf_reader = std.io.bufferedReader(file.reader());
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var in_stream = buf_reader.reader();
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var arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);
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defer arena.deinit();
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const allocator = &arena.allocator;
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var read_buf: [1024]u8 = undefined;
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const line = (try in_stream.readUntilDelimiterOrEof(&read_buf, '\n')).?;
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// e.g. "target area: x=20..30, y=-10..-5"
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var split = std.mem.split(line["target area: ".len..], ", ");
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var x_str = std.mem.split(split.next().?["x=".len..], "..");
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const x1 = try std.fmt.parseInt(i32, x_str.next().?, 10);
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const x2 = try std.fmt.parseInt(i32, x_str.next().?, 10);
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var y_str = std.mem.split(split.next().?["y=".len..], "..");
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const y1 = try std.fmt.parseInt(i32, y_str.next().?, 10);
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const y2 = try std.fmt.parseInt(i32, y_str.next().?, 10);
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// s: step
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// x: initial x velocity
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// y: initial y velocity
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// f(s,x) =
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// if (s < x) -(s-x)(s-x-1)/2 + x(x+1)/2
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// if (s >= x) x(x+1)/2
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// find max y such that there exists an s, there exists an x
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// y1 <= (-s^2 + 2sy + s)/2 <= y2
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// x1 <= f(s,x) <= x2
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// We hit target with y iff there exist a step s:
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// y1 <= -(s-y)(s-y-1)/2 + y*(y+1)/2 <= y2 <==>
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// y1 <= (-s*s + s*y + s + y*s)/2 <= y2 <==>
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// y1 <= (-s^2 + s*(2*y + 1))/2 <= y2 <==>
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// 0 <= -s^2 + s*(2*y + 1) - 2*y1
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// and 0 >= -s^2 + s*(2*y + 1) - 2*y2
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// solve for s:
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// y1s1 = (2*y + 1 - sqrt(4*y^2 + 4*y + 1 - 8*y1)) / 2
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// y1s2 = (2*y + 1 + sqrt(4*y^2 + 4*y + 1 - 8*y1)) / 2
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// y2s1 = (2*y + 1 - sqrt(4*y^2 + 4*y + 1 - 8*y2)) / 2
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// y2s1 = (2*y + 1 + sqrt(4*y^2 + 4*y + 1 - 8*y2)) / 2
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// y1s1 <= s <= y1s2 and (y2s2 <= s or s <= y2s1) <==>
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// max(y2s2,y1s1) <= s <= y1s2 or y1s1 <= s <= min(y1s2,y2s1)
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// Otherwise we cannot hit target with y:
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const highest_y: i32 = max(try abs(y1), try abs(y2));
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const lowest_y: i32 = 0;
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var y: i32 = highest_y;
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var s: i32 = undefined;
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y_loop: while (y >= lowest_y) : (y -= 1) {
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const y1s1 = (float(2 * y + 1) - sqrt(float(4 * y * y + 4 * y + 1 - 8 * y1))) / 2;
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const y1s2 = (float(2 * y + 1) + sqrt(float(4 * y * y + 4 * y + 1 - 8 * y1))) / 2;
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const y2s1 = (float(2 * y + 1) - sqrt(float(4 * y * y + 4 * y + 1 - 8 * y2))) / 2;
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const y2s2 = (float(2 * y + 1) + sqrt(float(4 * y * y + 4 * y + 1 - 8 * y2))) / 2;
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// iterate s
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s = max(max(ceil(y2s2), ceil(y1s1)), 0);
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while (s <= floor(y1s2)) : (s += 1) {
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if (xExists(y, s, x1, x2)) break :y_loop;
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}
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s = max(ceil(y1s1), 0);
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while (s <= min(floor(y1s2), floor(y2s1))) : (s += 1) {
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if (xExists(y, s, x1, x2)) break :y_loop;
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}
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}
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const stdout = std.io.getStdOut().writer();
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try stdout.print("{d}\n", .{y_s(y, y)});
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}
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fn y_s(y: i32, s: i32) i32 {
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return @divExact(-s * s + 2 * s * y + s, 2);
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}

‎day17/part2.zig‎

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const std = @import("std");
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const sqrt = std.math.sqrt;
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const abs = std.math.absInt;
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const max = std.math.max;
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const min = std.math.min;
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fn floor(x: f64) i32 {
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return @floatToInt(i32, @floor(x));
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}
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fn ceil(x: f64) i32 {
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return @floatToInt(i32, @ceil(x));
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}
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fn float(x: i32) f64 {
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return @intToFloat(f64, x);
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}
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fn divCeil(x: i32, y: i32) i32 {
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const res = @divFloor(x, y);
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return if (res * y == x) res else res + 1;
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}
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pub fn main() !void {
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var file = try std.fs.cwd().openFile("input", .{});
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defer file.close();
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var buf_reader = std.io.bufferedReader(file.reader());
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var in_stream = buf_reader.reader();
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var arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);
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defer arena.deinit();
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const allocator = &arena.allocator;
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var read_buf: [1024]u8 = undefined;
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const line = (try in_stream.readUntilDelimiterOrEof(&read_buf, '\n')).?;
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// e.g. "target area: x=20..30, y=-10..-5"
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var split = std.mem.split(line["target area: ".len..], ", ");
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var x_str = std.mem.split(split.next().?["x=".len..], "..");
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const x1 = try std.fmt.parseInt(i32, x_str.next().?, 10);
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const x2 = try std.fmt.parseInt(i32, x_str.next().?, 10);
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var y_str = std.mem.split(split.next().?["y=".len..], "..");
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const y1 = try std.fmt.parseInt(i32, y_str.next().?, 10);
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const y2 = try std.fmt.parseInt(i32, y_str.next().?, 10);
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// s: step
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// x: initial x velocity
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// y: initial y velocity
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// f(s,x) =
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// if (s < x) -(s-x)(s-x-1)/2 + x(x+1)/2
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// if (s >= x) x(x+1)/2
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// find max y such that there exists an s, there exists an x
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// y1 <= (-s^2 + 2sy + s)/2 <= y2
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// x1 <= f(s,x) <= x2
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// We hit target with y iff there exist a step s:
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// y1 <= -(s-y)(s-y-1)/2 + y*(y+1)/2 <= y2 <==>
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// y1 <= (-s*s + s*y + s + y*s)/2 <= y2 <==>
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// y1 <= (-s^2 + s*(2*y + 1))/2 <= y2 <==>
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// 0 <= -s^2 + s*(2*y + 1) - 2*y1
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// and 0 >= -s^2 + s*(2*y + 1) - 2*y2
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// solve for s:
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// y1s1 = (2*y + 1 - sqrt(4*y^2 + 4*y + 1 - 8*y1)) / 2
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// y1s2 = (2*y + 1 + sqrt(4*y^2 + 4*y + 1 - 8*y1)) / 2
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// y2s1 = (2*y + 1 - sqrt(4*y^2 + 4*y + 1 - 8*y2)) / 2
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// y2s1 = (2*y + 1 + sqrt(4*y^2 + 4*y + 1 - 8*y2)) / 2
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// y1s1 <= s <= y1s2 and (y2s2 <= s or s <= y2s1) <==>
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// max(y2s2,y1s1) <= s <= y1s2 or y1s1 <= s <= min(y1s2,y2s1)
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// Otherwise we cannot hit target with y:
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const highest_y: i32 = max(try abs(y1), try abs(y2));
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const lowest_y: i32 = -highest_y;
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var y: i32 = highest_y;
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var s: i32 = undefined;
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var num: u32 = 0;
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while (y >= lowest_y) : (y -= 1) {
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const y1s1 = (float(2 * y + 1) - sqrt(float(4 * y * y + 4 * y + 1 - 8 * y1))) / 2;
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const y1s2 = (float(2 * y + 1) + sqrt(float(4 * y * y + 4 * y + 1 - 8 * y1))) / 2;
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const y2s1 = (float(2 * y + 1) - sqrt(float(4 * y * y + 4 * y + 1 - 8 * y2))) / 2;
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const y2s2 = (float(2 * y + 1) + sqrt(float(4 * y * y + 4 * y + 1 - 8 * y2))) / 2;
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var xs = std.AutoHashMap(i32, void).init(allocator);
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defer xs.deinit();
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// iterate s
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s = max(max(ceil(y2s2), ceil(y1s1)), 0);
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while (s <= floor(y1s2)) : (s += 1) {
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const intervals = intervalX(y, s, x1, x2);
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for (intervals) |interval| {
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var x: i32 = interval.from;
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while (x <= interval.to) : (x += 1) {
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try xs.put(x, {});
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}
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}
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}
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s = max(ceil(y1s1), 0);
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while (s <= min(floor(y1s2), floor(y2s1))) : (s += 1) {
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const intervals = intervalX(y, s, x1, x2);
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for (intervals) |interval| {
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var x: i32 = interval.from;
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while (x <= interval.to) : (x += 1) {
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try xs.put(x, {});
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}
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}
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}
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num += xs.count();
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}
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const stdout = std.io.getStdOut().writer();
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try stdout.print("{d}\n", .{num});
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}
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const Interval = struct {
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from: i32,
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to: i32,
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};
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fn intervalX(y: i32, s: i32, x1: i32, x2: i32) [3]Interval {
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var intervals: [3]Interval = undefined;
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// Solve for x:
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// if s < x:
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// x1 <= sum x-s+1 .. x <= x2 <==>
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// x1 <= s*x - (s-1)*s/2 <= x2 <==>
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// x1/s + (s - 1)/2 <= x <= x2/s + (s - 1)/2
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var num: i32 = 0;
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{
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const min_x = max(divCeil(x1 + @divExact((s - 1) * s, 2), s), s + 1);
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const max_x = @divFloor(x2 + @divExact((s - 1) * s, 2), s);
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intervals[0] = Interval{ .from = min_x, .to = max_x };
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}
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// if s <= x:
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// x1 <= (x*x + x) / 2 <= x2
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// Solve:
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// sx1 = (-1 - sqrt(1 + 8*x1)) / 2
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// sx2 = (-1 + sqrt(1 + 8*x1)) / 2
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//
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// sx3 = (-1 - sqrt(1 + 8*x2)) / 2
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// sx4 = (-1 + sqrt(1 + 8*x2)) / 2
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//
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// so:
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// (x <= sx1 or sx2 <= x) and
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// (sx3 <= x <= sx4) <==>
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// (max(sx3, s) <= x <= min(sx1, sx4)) or
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// (max(sx2, sx3, s) <= x <= sx4)
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{
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const sx1 = floor((-1 - sqrt(float(1 + 8 * x1))) / 2);
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const sx2 = ceil((-1 + sqrt(float(1 + 8 * x1))) / 2);
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const sx3 = ceil((-1 - sqrt(float(1 + 8 * x2))) / 2);
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const sx4 = floor((-1 + sqrt(float(1 + 8 * x2))) / 2);
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{
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const min_x = max(sx3, 0);
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const max_x = min(min(sx1, sx4), s);
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intervals[1] = Interval{ .from = min_x, .to = max_x };
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}
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{
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const min_x = max(max(sx2, sx3), 0);
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const max_x = min(sx4, s);
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intervals[2] = Interval{ .from = min_x, .to = max_x };
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}
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}
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return intervals;
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}

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