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Five Fives (Posted on 2005-09-18) Difficulty: 4 of 5
A mathematician who was exceedingly fond of the number five set to work trying to express as many consecutive integers using no numerals besides '5', and only up to five of them. She allowed herself to use any standard mathematical notation she knew, as long as it didn't contain any numerals. For example, she could use the symbol for 'square root', but not 'cube root' (because it contains a '3'). She determined that the highest consecutive integer she could express this way was 36. Her last few calculations were as follows:
  • 31 = 5*5 + 5 + (5/5)
  • 32 = 55*.5 + 5 - .5
  • 33 = (55 + 5) * .55
  • 34 = 5!/5 + 5/.5
  • 35 = (5 + (5+5)/5) * 5
  • 36 = 5*5 + 55/5
  • 37 = ?
Was she correct in thinking 36 was the highest consecutive integer she could express this way? Can you express 37 using only up to five 5's?

Note: The intention here is to find an exact expression, so rounding expressions like [] "greatest integer" are not allowed.
Note: Can you do it without using letters of any kind (x, log, lim, sum, etc.)?

See The Solution Submitted by Josh70679    
Rating: 4.4737 (19 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution more | Comment 267 of 395 |
<TABLE cellSpacing=0 cellPadding=2 width=336 border=0> <TBODY> <TR> <TD width="20%" height=16>

432

</TD> <TD width="20%" height=16>

400

</TD> <TD width="20%" height=16>

385

</TD> <TD width="20%" height=16>

360

</TD> <TD width="20%" height=16>

</TD></TR> <TR> <TD width="20%" height=16>

431

</TD> <TD width="20%" height=16>

399

</TD> <TD width="20%" height=16>

384

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

1

</TD></TR> <TR> <TD width="20%" height=16>

429

</TD> <TD width="20%" height=16>

397

</TD> <TD width="20%" height=16>

381

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

3

</TD></TR> <TR> <TD width="20%" height=16>

427

</TD> <TD width="20%" height=16>

395

</TD> <TD width="20%" height=16>

376

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

5

</TD></TR> <TR> <TD width="20%" height=16>

426

</TD> <TD width="20%" height=16>

394

</TD> <TD width="20%" height=16>

370

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

6

</TD></TR> <TR> <TD width="20%" height=16>

425

</TD> <TD width="20%" height=16>

393

</TD> <TD width="20%" height=16>

363

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

7

</TD></TR> <TR> <TD width="20%" height=16>

423

</TD> <TD width="20%" height=16>

391

</TD> <TD width="20%" height=16>

354

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

9

</TD></TR> <TR> <TD width="20%" height=16>

422

</TD> <TD width="20%" height=16>

390

</TD> <TD width="20%" height=16>

344

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

10

</TD></TR> <TR> <TD width="20%" height=16>

407

</TD> <TD width="20%" height=16>

375

</TD> <TD width="20%" height=16>

319

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

25

</TD></TR> <TR> <TD width="20%" height=16>

400

</TD> <TD width="20%" height=16>

368

</TD> <TD width="20%" height=16>

287

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

32

</TD></TR> <TR> <TD width="20%" height=16>

377

</TD> <TD width="20%" height=16>

345

</TD> <TD width="20%" height=16>

232

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

55

</TD></TR> <TR> <TD width="20%" height=16>

372

</TD> <TD width="20%" height=16>

340

</TD> <TD width="20%" height=16>

172

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

60

</TD></TR> <TR> <TD width="20%" height=16>

366

</TD> <TD width="20%" height=16>

334

</TD> <TD width="20%" height=16>

106

</TD> <TD width="20%" height=16>

</TD> <TD width="20%" height=16>

66

</TD></TR> <TR> <TD width="20%" height=16>

433

</TD> <TD width="20%" height=16>

401

</TD> <TD width="20%" height=16>

386

</TD> <TD width="20%" height=16>

361

</TD> <TD width="20%" height=16>

1

</TD></TR> <TR> <TD width="20%" height=16>

435

</TD> <TD width="20%" height=16>

403

</TD> <TD width="20%" height=16>

388

</TD> <TD width="20%" height=16>

363

</TD> <TD width="20%" height=16>

3

</TD></TR> <TR> <TD width="20%" height=16>

437

</TD> <TD width="20%" height=16>

405

</TD> <TD width="20%" height=16>

390

</TD> <TD width="20%" height=16>

365

</TD> <TD width="20%" height=16>

5

</TD></TR> <TR> <TD width="20%" height=16>

438

</TD> <TD width="20%" height=16>

406

</TD> <TD width="20%" height=16>

391

</TD> <TD width="20%" height=16>

366

</TD> <TD width="20%" height=16>

6

</TD></TR> <TR> <TD width="20%" height=16>

439

</TD> <TD width="20%" height=16>

407

</TD> <TD width="20%" height=16>

392

</TD> <TD width="20%" height=16>

367

</TD> <TD width="20%" height=16>

7

</TD></TR> <TR> <TD width="20%" height=16>

441

</TD> <TD width="20%" height=16>

409

</TD> <TD width="20%" height=16>

394

</TD> <TD width="20%" height=16>

369

</TD> <TD width="20%" height=16>

9

</TD></TR> <TR> <TD width="20%" height=16>

442

</TD> <TD width="20%" height=16>

410

</TD> <TD width="20%" height=16>

395

</TD> <TD width="20%" height=16>

370

</TD> <TD width="20%" height=16>

10

</TD></TR> <TR> <TD width="20%" height=16>

457

</TD> <TD width="20%" height=16>

425

</TD> <TD width="20%" height=16>

410

</TD> <TD width="20%" height=16>

385

</TD> <TD width="20%" height=16>

25

</TD></TR> <TR> <TD width="20%" height=16>

464

</TD> <TD width="20%" height=16>

432

</TD> <TD width="20%" height=16>

417

</TD> <TD width="20%" height=16>

392

</TD> <TD width="20%" height=16>

32

</TD></TR> <TR> <TD width="20%" height=16>

487

</TD> <TD width="20%" height=16>

455

</TD> <TD width="20%" height=16>

440

</TD> <TD width="20%" height=16>

415

</TD> <TD width="20%" height=16>

55

</TD></TR> <TR> <TD width="20%" height=16>

492

</TD> <TD width="20%" height=16>

460

</TD> <TD width="20%" height=16>

445

</TD> <TD width="20%" height=16>

420

</TD> <TD width="20%" height=16>

60

</TD></TR> <TR> <TD width="20%" height=16>

498

</TD> <TD width="20%" height=16>

466

</TD> <TD width="20%" height=16>

451

</TD> <TD width="20%" height=16>

426

</TD> <TD width="20%" height=16>

66

</TD></TR></TBODY></TABLE>

This table is self explanatory for the " involved"

it is easy to extract the "missing in action" like

368,371 etc   

JosH ,  your  list is unprintable, except the 1st page.


  Posted by Ady TZIDON on 2005-10-14 08:14:37
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