perplexus.info :: Numbers : Who's on the 4th base?

Who's on the 4th base? (Posted on 2019-05-28)

21 is a palindrome in 4 distinct bases (bases: 2,4,6,20 ==> 10101,111,33,11,)

Find the smallest 3-digit number N such that both this number and its reversal are palindromes in at least 4 distinct bases.

Both number 1 and numbers higher than N do not qualify as bases.
Base 10 is in.

No Solution Yet Submitted by Ady TZIDON

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re: computer solution and extension continued

| Comment 2 of 3 |

(In reply to computer solution and extension by Charlie)

545

4 10 5 4 5 17 1 15 1 108 5 5 544 1 1

546

11 4 2 0 2 0 2 9 6 6 6 16 2 2 2 25 21 21 38 14 14 41 13 13 77 7 7 90 6 6 181 3 3 272 2 2 545 1 1

6 2 1 0 1 0 0 0 0 1 0 1 23 1 5 1 42 15 15 128 5 5 214 3 3 644 1 1

548

4 14 2 11 2 136 4 4 273 2 2 547 1 1

4 12 5 10 5 64 13 13 168 5 5 844 1 1

549

4 13 3 3 3 60 9 9 182 3 3 548 1 1

9 8 1 6 6 1 34 27 27 44 21 21 62 15 15 104 9 9 134 7 7 188 5 5 314 3 3 944 1 1

555

7 9 6 7 6 10 5 5 5 12 3 10 3 36 15 15 110 5 5 184 3 3 554 1 1

558

6 30 18 18 61 9 9 92 6 6 185 3 3 278 2 2 557 1 1

7 4 3 1 1 1 3 44 19 19 56 15 15 94 9 9 170 5 5 284 3 3 854 1 1

565

4 10 5 6 5 11 4 7 4 112 5 5 564 1 1

567

6 12 3 11 3 26 21 21 62 9 9 80 7 7 188 3 3 566 1 1

8 2 1 0 1 1 1 1 1 1 0 1 11 6 3 6 44 17 17 50 15 15 84 9 9 152 5 5 254 3 3 764 1 1

568

6 7 1 4 4 1 21 1 6 1 70 8 8 141 4 4 283 2 2 567 1 1

5 11 7 1 7 24 1 12 1 27 1 5 1 172 5 5 864 1 1

575

5 10 5 7 5 13 3 5 3 24 23 23 114 5 5 574 1 1

576

13 11 4 8 4 14 2 13 2 23 1 2 1 31 18 18 35 16 16 47 12 12 63 9 9 71 8 8 95 6 6 143 4 4 191 3 3 287 2 2 575 1 1

7 14 3 6 3 26 25 25 44 15 15 74 9 9 134 5 5 224 3 3 674 1 1

578

4 16 2 4 2 33 17 17 288 2 2 577 1 1

5 23 1 15 1 34 25 25 124 7 7 174 5 5 874 1 1

585

9 2 1 0 0 1 0 0 1 0 0 1 8 1 1 1 1 10 5 8 5 38 15 15 44 13 13 64 9 9 116 5 5 194 3 3 584 1 1

588

10 13 3 6 3 27 21 21 41 14 14 48 12 12 83 7 7 97 6 6 146 4 4 195 3 3 293 2 2 587 1 1

5 26 1 8 1 58 15 15 176 5 5 294 3 3 884 1 1

595

7 10 5 9 5 18 1 15 1 22 1 5 1 34 17 17 84 7 7 118 5 5 594 1 1

606

5 10 6 0 6 100 6 6 201 3 3 302 2 2 605 1 1

608

7 3 2 1 1 1 1 2 31 19 19 37 16 16 75 8 8 151 4 4 303 2 2 607 1 1

6 5 1 1 2 1 1 23 1 12 1 30 26 26 61 13 13 402 2 2 805 1 1

609

5 19 1 13 1 28 21 21 86 7 7 202 3 3 608 1 1

5 12 6 3 6 150 6 6 301 3 3 452 2 2 905 1 1

616

10 10 6 1 6 12 4 3 4 27 22 22 43 14 14 55 11 11 76 8 8 87 7 7 153 4 4 307 2 2 615 1 1

618

4 102 6 6 205 3 3 308 2 2 617 1 1

11 14 4 2 4 33 24 24 47 17 17 50 16 16 67 12 12 101 8 8 135 6 6 203 4 4 271 3 3 407 2 2 815 1 1

626

6 5 1 0 0 0 1 10 6 2 6 16 2 7 2 25 1 0 1 312 2 2 625 1 1

627

5 13 3 9 3 32 19 19 56 11 11 208 3 3 626 1 1

8 5 1 0 4 0 1 25 1 4 1 32 22 22 65 11 11 120 6 6 241 3 3 362 2 2 725 1 1

628

6 9 7 6 7 12 4 4 4 19 1 14 1 156 4 4 313 2 2 627 1 1

5 25 1 8 1 58 14 14 117 7 7 412 2 2 825 1 1

636

7 10 6 3 6 52 12 12 105 6 6 158 4 4 211 3 3 317 2 2 635 1 1

637

4 9 7 7 7 48 13 13 90 7 7 636 1 1

7 21 1 14 1 31 23 23 45 16 16 91 8 8 183 4 4 367 2 2 735 1 1

638

4 28 22 22 57 11 11 318 2 2 637 1 1

7 13 4 12 4 37 22 22 43 19 19 75 11 11 208 4 4 417 2 2 835 1 1

639

4 22 1 7 1 70 9 9 212 3 3 638 1 1

13 5 1 2 2 2 1 35 26 26 38 24 24 51 18 18 71 13 13 77 12 12 103 9 9 116 8 8 155 6 6 233 4 4 311 3 3 467 2 2 935 1 1

646

6 9 7 8 7 10 6 4 6 33 19 19 37 17 17 322 2 2 645 1 1

648

11 17 2 4 2 26 24 24 35 18 18 53 12 12 71 9 9 80 8 8 107 6 6 161 4 4 215 3 3 323 2 2 647 1 1

6 46 18 18 93 9 9 140 6 6 281 3 3 422 2 2 845 1 1

656

8 3 2 2 0 0 2 2 9 8 0 8 10 6 5 6 40 16 16 81 8 8 163 4 4 327 2 2 655 1 1

657

4 8 1 2 2 1 72 9 9 218 3 3 656 1 1

12 5 1 1 0 1 1 35 21 21 41 18 18 53 14 14 62 12 12 83 9 9 107 7 7 125 6 6 188 4 4 251 3 3 377 2 2 755 1 1

658

5 16 2 9 2 46 14 14 93 7 7 328 2 2 657 1 1

6 5 1 1 4 1 1 7 2 3 3 2 106 8 8 213 4 4 427 2 2 855 1 1

666

10 4 2 2 1 2 2 10 6 6 6 13 3 12 3 19 1 16 1 36 18 18 73 9 9 110 6 6 221 3 3 332 2 2 665 1 1

676

9 5 1 0 2 0 1 10 6 7 6 11 5 6 5 12 4 8 4 25 1 2 1 51 13 13 168 4 4 337 2 2 675 1 1

678

5 15 3 0 3 112 6 6 225 3 3 338 2 2 677 1 1

10 11 7 2 7 13 5 2 5 19 2 8 2 25 1 10 1 72 12 12 145 6 6 218 4 4 291 3 3 437 2 2 875 1 1

686

6 10 6 8 6 18 2 2 2 48 14 14 97 7 7 342 2 2 685 1 1

688

7 7 2 0 0 2 12 4 9 4 42 16 16 85 8 8 171 4 4 343 2 2 687 1 1

4 5 1 2 0 2 1 14 4 7 4 442 2 2 885 1 1

696

9 10 6 9 6 28 24 24 57 12 12 86 8 8 115 6 6 173 4 4 231 3 3 347 2 2 695 1 1

698

4 4 2 2 3 2 2 11 5 8 5 348 2 2 697 1 1

8 31 28 28 55 16 16 63 14 14 111 8 8 127 7 7 223 4 4 447 2 2 895 1 1

717

5 2 1 0 1 1 0 0 1 1 0 1 10 7 1 7 14 3 9 3 238 3 3 716 1 1

737

5 10 7 3 7 12 5 1 5 23 1 9 1 66 11 11 736 1 1

738

8 15 3 4 3 16 2 14 2 40 18 18 81 9 9 122 6 6 245 3 3 368 2 2 737 1 1

5 22 1 16 1 30 27 27 92 9 9 278 3 3 836 1 1

747

4 10 7 4 7 82 9 9 248 3 3 746 1 1

748

6 33 22 22 43 17 17 67 11 11 186 4 4 373 2 2 747 1 1

4 3 1 0 1 1 1 0 1 76 11 11 120 7 7 846 1 1

749

4 12 5 2 5 22 1 12 1 106 7 7 748 1 1

4 16 3 11 3 21 2 3 2 22 1 21 1 946 1 1

757

5 3 1 0 0 1 0 0 1 10 7 5 7 21 1 15 1 27 1 1 1 756 1 1

759

5 14 3 12 3 32 23 23 68 11 11 252 3 3 758 1 1

4 32 29 29 86 11 11 318 3 3 956 1 1

767

4 10 7 6 7 17 2 11 2 58 13 13 766 1 1

768

10 15 3 6 3 31 24 24 47 16 16 63 12 12 95 8 8 127 6 6 191 4 4 255 3 3 383 2 2 767 1 1

4 16 3 6 3 50 17 17 288 3 3 866 1 1

777

6 6 3 3 3 3 10 7 7 7 36 21 21 110 7 7 258 3 3 776 1 1

787

5 4 3 0 1 0 3 10 7 8 7 11 6 5 6 16 3 1 3 786 1 1

788

4 14 4 0 4 196 4 4 393 2 2 787 1 1

4 4 3 1 3 1 3 11 7 3 7 17 3 1 3 886 1 1

797

4 10 7 9 7 12 5 6 5 13 4 9 4 796 1 1

798

10 11 6 6 6 15 3 8 3 37 21 21 41 19 19 56 14 14 113 7 7 132 6 6 265 3 3 398 2 2 797 1 1

5 28 1 4 1 38 23 23 68 13 13 298 3 3 896 1 1

808

5 10 8 0 8 100 8 8 201 4 4 403 2 2 807 1 1

818

4 10 8 1 8 17 2 14 2 408 2 2 817 1 1

819

11 2 1 1 0 0 1 1 0 0 1 1 4 3 0 3 0 3 6 3 4 4 3 16 3 3 3 19 2 5 2 38 21 21 62 13 13 90 9 9 116 7 7 272 3 3 818 1 1

9 12 6 4 6 33 27 27 50 18 18 53 17 17 101 9 9 152 6 6 305 3 3 458 2 2 917 1 1

828

11 10 8 2 8 15 3 10 3 35 23 23 45 18 18 68 12 12 91 9 9 137 6 6 206 4 4 275 3 3 413 2 2 827 1 1

838

5 10 8 3 8 19 2 6 2 27 1 4 1 418 2 2 837 1 1

848

7 10 8 4 8 18 2 11 2 52 16 16 105 8 8 211 4 4 423 2 2 847 1 1

858

11 10 8 5 8 14 4 5 4 15 3 12 3 32 26 26 38 22 22 65 13 13 77 11 11 142 6 6 285 3 3 428 2 2 857 1 1

868

8 6 4 0 0 4 10 8 6 8 30 28 28 61 14 14 123 7 7 216 4 4 433 2 2 867 1 1

888

10 10 8 8 8 15 3 14 3 36 24 24 73 12 12 110 8 8 147 6 6 221 4 4 295 3 3 443 2 2 887 1 1

889

4 13 5 3 5 24 1 13 1 126 7 7 888 1 1

6 37 26 26 51 19 19 75 13 13 246 4 4 493 2 2 987 1 1

898

5 10 8 9 8 11 7 4 7 23 1 16 1 448 2 2 897 1 1

909

5 10 9 0 9 11 7 5 7 100 9 9 302 3 3 908 1 1

919

4 10 9 1 9 15 4 1 4 27 1 7 1 918 1 1

939

4 4 3 2 2 2 3 10 9 3 9 312 3 3 938 1 1

949

4 10 9 4 9 15 4 3 4 72 13 13 948 1 1

969

5 10 9 6 9 50 19 19 56 17 17 322 3 3 968 1 1

979

5 10 9 7 9 15 4 5 4 16 3 13 3 88 11 11 978 1 1

989

6 10 9 8 9 17 3 7 3 21 2 5 2 26 1 12 1 42 23 23 988 1 1

999

6 10 9 9 9 14 5 1 5 36 27 27 110 9 9 332 3 3 998 1 1

208 done

DefDbl A-Z

Dim crlf$, dig(10), psw

Private Sub Form_Load()

Form1.Visible = True

Text1.Text = ""

crlf = Chr(13) + Chr(10)

For n = 100 To 999

ns$ = LTrim(Str(n))

If Left(ns, 1) <= Right(ns, 1) Then

If numpals(ns) >= 4 Then

r$ = ""

For i = 1 To Len(ns)

r = Mid(ns, i, 1) + r

r = LTrim(Str(Val(r)))

If numpals(r) >= 4 Then

Text1.Text = Text1.Text & n & crlf

ansct = ansct + 1

Text1.Text = Text1.Text & numpals(ns)

psw = 1

np = numpals(ns)

Text1.Text = Text1.Text & " " & crlf

psw = 0

Text1.Text = Text1.Text & numpals(r)

psw = 1

np = numpals(r)

Text1.Text = Text1.Text & " " & crlf & crlf

psw = 0

End If

DoEvents

Text1.Text = Text1.Text & crlf & ansct & " done"

End Sub

Function numpals(ss$)

s = Val(ss)

ct = 0

For b = 2 To s

numdig = base(s, b)

good = 1

For i = 1 To numdig / 2

If dig(i) <> dig(numdig + 1 - i) Then good = 0: Exit For

If good Then

ct = ct + 1

If psw Then

Text1.Text = Text1.Text & " " & b & " "

For i = numdig To 1 Step -1

Text1.Text = Text1.Text & Str(dig(i))

End If

numpals = ct

End Function

Function base(n, b)

v$ = ""

n2 = n

q = Int(n2 / b)

d = n2 - q * b

n2 = q

psn = psn + 1

dig(psn) = d

Loop Until n2 = 0

base = psn

End Function

Posted by Charlie on 2019-05-28 15:53:51

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