<< problem 150 - Searching a triangular array for a sub-triangle ... Writing 1/2 as a sum of inverse squares - problem 152 >>

# Problem 151: Paper sheets of standard sizes: an expected-value problem

A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5.

Every Monday morning, the foreman opens a new envelope, containing a large sheet of the special paper with size A1.

He proceeds to cut it in half, thus getting two sheets of size A2. Then he cuts one of them in half to get two sheets of size A3
and so on until he obtains the A5-size sheet needed for the first batch of the week.

All the unused sheets are placed back in the envelope.

At the beginning of each subsequent batch, he takes from the envelope one sheet of paper at random.
If it is of size A5, he uses it. If it is larger, he repeats the 'cut-in-half' procedure until he has what he needs and any remaining sheets are always placed back in the envelope.

Excluding the first and last batch of the week, find the expected number of times (during each week) that the foreman finds a single sheet of paper in the envelope.

Give your answer rounded to six decimal places using the format x.xxxxxx .

# My Algorithm

I solved this problem twice. My first attempt was running a Monte-Carlo simulation - and never achieved enough precision. That code is still present below.

My second (and successful) attempt evaluates the exact probabilities of each possible combination.
Its parameter is a simple container with 5 elements, where the first represents the number of A1 sheets, the second element stands for the number of A2 sheets, etc.
The initial parameter is { 1, 0, 0, 0, 0 }. If at some point we have no A5, no A4, 2x A3, no A4 and 1x A5 then it would be { 0, 0, 2, 0, 1 }.

Each sheet size is picked and the function calls itself recursively. The probability for picking a single sheet is 1 / numSheets.
If there are multiple sheets of the same sheet size i then the probability increases to sheets[i] / numSheets.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

       #include <iostream>
#include <vector>
#include <iomanip>

// return number of batch where a single sheet is in the envelope
// parameter is a list of the number of available sheets, starting with A1
// e.g. 0x A5, 0x A4, 2x A3, 0x A4, 1x A5 is encoded as { 0, 0, 2, 0, 1 }
double evaluate(std::vector<unsigned int> sheets)
{
// count sheets
unsigned int numSheets = 0;
for (auto s : sheets)
numSheets += s;

// a single sheet ?
double single = 0;
if (numSheets == 1)
{
// only one sheet of A5 left => last batch
if (sheets.back() == 1)
return 0;

// except if the single sheet is A1 (first batch)
if (sheets.front() == 0)
single = 1;
}

// process all sheet sizes
for (size_t i = 0; i < sheets.size(); i++)
{
if (sheets[i] == 0)
continue;

auto next = sheets;
// use one piece of the sheet size
next[i]--;
// cut it into smaller pieces
for (size_t j = i + 1; j < next.size(); j++)
next[j]++;

// how likely do we pick this sheet size ?
double probability = sheets[i] / (double)numSheets;
// analyze next batch
single += evaluate(next) * probability;
}

return single;
}

// I kept this code for historical reasons ... it's not called in main()
double montecarlo()
{
// more rounds improve precision
const unsigned int NumRounds = 1000000;
// different seeds yields different results ...
srand(111);

// how often a single sheet is observed
unsigned int singleSheet = 0;
for (unsigned int round = 0; round < NumRounds; round++)
{
// 1 => DIN A1, 2 => A2, ... 5 => A5
// this stack of sheets may contain some sizes multiple times
const unsigned int SheetSizes = 5;
unsigned int sheets[SheetSizes] = { 1,0,0,0,0 }; // one sheet A1 on Monday morning
unsigned int numSheets = 1;

// until all sheets are used (on Friday afternoon)
while (numSheets > 0)
{
// a single sheet ?
if (numSheets == 1)
singleSheet++;

// pick a random sheet
unsigned int pick = rand() % numSheets;
unsigned int current = 0;
// select sheet size
while (pick >= sheets[current])
pick -= sheets[current++];
// and remove one sheet
sheets[current]--;

// reduce total number of sheets, too
numSheets--;

// if the current sheet is larger than A5 then cut it into smaller sheets
while (++current < SheetSizes)
{
sheets[current]++;
numSheets++;
}
}

// don't count the first and last batch (always one sheet)
singleSheet -= 2;
}

return singleSheet / (double)NumRounds;
}

int main()
{
std::cout << std::fixed << std::setprecision(6);

std::cout << evaluate({ 1,0,0,0,0 }) << std::endl;

// my first approach was using a Monte-Carlo simulation but it converges too slowly
//std::cout << montecarlo() << std::endl;

return 0;
}


This solution contains 19 empty lines, 29 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.02 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

May 23, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 35% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until a new problem is published the flashing problem is the one I solved most recently

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500
 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
 601 602 603 604 605 606 607 608 609 610 611
The 286 solved problems (level 11) had an average difficulty of 31.8% at Project Euler and
I scored 13,486 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

 << problem 150 - Searching a triangular array for a sub-triangle ... Writing 1/2 as a sum of inverse squares - problem 152 >>
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !