List all the numbers (below 4000) that their Roman numeral representations have the same number of letters as their squares.
(In reply to
computer solution not considering overlining for large squares by Charlie)
Adding code to allow for overbarred numerals, the list expands considerably. Fortunately, 1000, either as M or an overbarred I, still is one character so it doesn't affect the count. In the below, for the squares of 32 - 39, the same numbers appear below in a different manner, but the same character count. I've put the overbarred letters on the left, followed by a space and then the non-barred letters, but the space doesn't count in the character count. In some cases there is/are only overbarred letters; so if there is no space and the square is over 1000, be assured all the letters should be overbarred, such as X for 10000, the square of 100 (C) or M for 1,000,000, the square of M for 1,000.
1 1 I I
2 4 II IV
8 64 VIII LXIV
10 100 X C
20 400 XX CD
23 529 XXIII DXXIX
32 1024 XXXII I XXIV
34 1156 XXXIV I CLVI
38 1444 XXXVIII I CDXLIV
39 1521 XXXIX I DXXI
71 5041 LXXI V XLI
80 6400 LXXX VI CD
84 7056 LXXXIV VII LVI
98 9604 XCVIII IX DCIV
100 10000 C X
148 21904 CXLVIII XXI CMIV
200 40000 CC XL
227 51529 CCXXVII LI DXXIX
230 52900 CCXXX LII CM
237 56169 CCXXXVII LVI CLXIX
238 56644 CCXXXVIII LVI DCXLIV
243 59049 CCXLIII LIX XLIX
245 60025 CCXLV LX XXV
246 60516 CCXLVI LX DXVI
253 64009 CCLIII LXIV IX
320 102400 CCCXX CII CD
331 109561 CCCXXXI CIX DLXI
332 110224 CCCXXXII CX CCXXIV
338 114244 CCCXXXVIII CXIV CCXLIV
340 115600 CCCXL CXV DC
380 144400 CCCLXXX CXLIV CD
381 145161 CCCLXXXI CXLV CLXI
389 151321 CCCLXXXIX CLI CCCXXI
390 152100 CCCXC CLII C
399 159201 CCCXCIX CLIX CCI
436 190096 CDXXXVI CXC XCVI
449 201601 CDXLIX CCI DCI
638 407044 DCXXXVIII CDVII XLIV
710 504100 DCCX DIV C
718 515524 DCCXVIII DXV DXXIV
742 550564 DCCXLII DL DLXIV
743 552049 DCCXLIII DLII XLIX
745 555025 DCCXLV DLV XXV
747 558009 DCCXLVII DLVIII IX
775 600625 DCCLXXV DC DCXXV
782 611524 DCCLXXXII DCXI DXXIV
800 640000 DCCC DCXL
831 690561 DCCCXXXI DCXC DLXI
834 695556 DCCCXXXIV DCXCV DLVI
838 702244 DCCCXXXVIII DCCII CCXLIV
840 705600 DCCCXL DCCV DC
898 806404 DCCCXCVIII DCCCVI CDIV
980 960400 CMLXXX CMLX CD
998 996004 CMXCVIII CMXCVI IV
1000 1000000 M M
1049 1100401 MXLIX MC CDI
1227 1505529 MCCXXVII MDV DXXIX
1245 1550025 MCCXLV MDL XXV
1248 1557504 MCCXLVIII MDLVII DIV
1253 1570009 MCCLIII MDLXX IX
1379 1901641 MCCCLXXIX MCMI DCXLI
1384 1915456 MCCCLXXXIV MCMXV CDLVI
1388 1926544 MCCCLXXXVIII MCMXXVI DXLIV
1397 1951609 MCCCXCVII MCMLI DCIX
1398 1954404 MCCCXCVIII MCMLIV CDIV
1416 2005056 MCDXVI MMV LVI
1468 2155024 MCDLXVIII MMCLV XXIV
1480 2190400 MCDLXXX MMCXC CD
1584 2509056 MDLXXXIV MMDIX LVI
1738 3020644 MDCCXXXVIII MMMXX DCXLIV
1871 3500641 MDCCCLXXI MMMD DCXLI
1888 3564544 MDCCCLXXXVII MMMDLXIV DXLIV
1898 3602404 MDCCCXCVIII MMMDCII CDIV
DefDbl A-Z
Dim crlf$
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Function pad$(x$, l)
pad$ = Left(x + Space$(l), l)
End Function
Private Sub Form_Load()
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents
For i = 1 To 4000
rm$ = roman(i)
l = Len(rm$)
sq = i * i
If sq > 1000 Then
pt1 = sq \ 1000: pt2 = sq Mod 1000
pt1r$ = roman(pt1): pt2r$ = roman(pt2)
l2 = Len(pt1r + pt2r)
rmsq$ = pt1r + " " + pt2r
Else
l2 = Len(roman(sq))
rmsq$ = roman(sq)
End If
If l = l2 Then
Text1.Text = Text1.Text & mform(i, "#####0") & mform(i * i, "########0") & " "
Text1.Text = Text1.Text & pad(rm$, 12) & " " & rmsq & crlf
End If
DoEvents
Next
Text1.Text = Text1.Text & "done" & crlf
End Sub
Function roman$(n)
q = n \ 1000: r = n Mod 1000
ro$ = String$(q, "M")
n2 = r
q = n2 \ 100: r = n2 Mod 100
Select Case q
Case 9: ro$ = ro$ + "CM"
Case 5 To 8: ro$ = ro$ + "D" + String$(q - 5, "C")
Case 4: ro$ = ro$ + "CD"
Case 0 To 3: ro$ = ro$ + String$(q, "C")
End Select
n2 = r
q = n2 \ 10: r = n2 Mod 10
Select Case q
Case 9: ro$ = ro$ + "XC"
Case 5 To 8: ro$ = ro$ + "L" + String$(q - 5, "X")
Case 4: ro$ = ro$ + "XL"
Case 0 To 3: ro$ = ro$ + String$(q, "X")
End Select
n2 = r
q = n2
Select Case q
Case 9: ro$ = ro$ + "IX"
Case 5 To 8: ro$ = ro$ + "V" + String$(q - 5, "I")
Case 4: ro$ = ro$ + "IV"
Case 0 To 3: ro$ = ro$ + String$(q, "I")
End Select
roman$ = ro$
End Function
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Posted by Charlie
on 2015-05-22 13:05:09 |
adding overbars for large squares
