A parallelogram has coordinates (0, 0), (144, 0), (377, 89) and (233, 89).
Determine the number of lattice points strictly within the interior of the parallelogram.
The four sides are segments of the following four equations:
y = 0
y = 89
x = (233/89)y
x = (233/89)y + 144
So we can count the lattice points by varying y from 1 to 88, plugging into the two equations for x and count the integers between the two x values. When y values are integers, none of the x values will be integers. So the number of integers between the two x values will always be 144.
So the number of lattice points on the interior is 144*88 = 12672.
If you want to include the border lines (which we do not since the puzzle asks for the interior), add 290 (145 each for the horizontal segments, and none for the oblique segments).
https://www.desmos.com/calculator/6kfdyed6xo
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Verification that there are 144 lattice points on each horizontal line, which isn't really necessary but it's an excuse to write a small program:
for y in range(1,89):
x1 = (233/89)*y
x2 = (233/89)*y + 144
print(1+int(x1), int(x2), int(x2) - (int(x1)))
3 146 144
6 149 144
8 151 144
11 154 144
14 157 144
16 159 144
19 162 144
21 164 144
24 167 144
27 170 144
29 172 144
32 175 144
35 178 144
37 180 144
40 183 144
42 185 144
45 188 144
48 191 144
50 193 144
53 196 144
55 198 144
58 201 144
61 204 144
63 206 144
66 209 144
69 212 144
71 214 144
74 217 144
76 219 144
79 222 144
82 225 144
84 227 144
87 230 144
90 233 144
92 235 144
95 238 144
97 240 144
100 243 144
103 246 144
105 248 144
108 251 144
110 253 144
113 256 144
116 259 144
118 261 144
121 264 144
124 267 144
126 269 144
129 272 144
131 274 144
134 277 144
137 280 144
139 282 144
142 285 144
144 287 144
147 290 144
150 293 144
152 295 144
155 298 144
158 301 144
160 303 144
163 306 144
165 308 144
168 311 144
171 314 144
173 316 144
176 319 144
179 322 144
181 324 144
184 327 144
186 329 144
189 332 144
192 335 144
194 337 144
197 340 144
199 342 144
202 345 144
205 348 144
207 350 144
210 353 144
213 356 144
215 358 144
218 361 144
220 363 144
223 366 144
226 369 144
228 371 144
231 374 144
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Posted by Larry
on 2024-12-21 08:20:46 |
solution
