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Thread: Right Triangles

  1. #1
    thinBasic MVPs danbaron's Avatar
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    Right Triangles

    For a right triangle, the Pythagorean theorem says that,

    a^2 + b^2 = c^2.

    If I am correct, here are all of the 1034 right triangles, for integer values of a, b, and c, such that a and b are not greater than 1000.



    ' code --------------------------------------------------------------------------------------------------------------------------------------
    
    #lang racket
    
    (define list-list (list))
    
    (define (list-integer-right-triangles n)
      (define a 0)
      (define b 0)
      (define c 0)
      (define count1 0)
      (define count2 0)
      (define (loop1)
        (set! count1 (add1 count1))
        (cond
          ((> count1 n) list-list)
          (else       
           (set! count2 (- count1 1))
           (loop2)
           (loop1))))
      (define (loop2)
        (set! count2 (add1 count2))
        (cond
          ((> count2 n))
          (else
           (set! a count1)
           (set! b count2)
           (set! c (sqrt (+ (expt a 2) (expt b 2))))
           (when (= c (floor c)) (set! list-list (append list-list (list (list a b c)))))
           (loop2))))
      (loop1))
    
    ' REPL interactions -------------------------------------------------------------------------------------------------------------------------
    
    Welcome to DrRacket, version 5.1 [3m].
    Language: racket.
    
    > (list-integer-right-triangles 1000)
    '((3 4 5)
      (5 12 13)
      (6 8 10)
      (7 24 25)
      (8 15 17)
      (9 12 15)
      (9 40 41)
      (10 24 26)
      (11 60 61)
      (12 16 20)
      (12 35 37)
      (13 84 85)
      (14 48 50)
      (15 20 25)
      (15 36 39)
      (15 112 113)
      (16 30 34)
      (16 63 65)
      (17 144 145)
      (18 24 30)
      (18 80 82)
      (19 180 181)
      (20 21 29)
      (20 48 52)
      (20 99 101)
      (21 28 35)
      (21 72 75)
      (21 220 221)
      (22 120 122)
      (23 264 265)
      (24 32 40)
      (24 45 51)
      (24 70 74)
      (24 143 145)
      (25 60 65)
      (25 312 313)
      (26 168 170)
      (27 36 45)
      (27 120 123)
      (27 364 365)
      (28 45 53)
      (28 96 100)
      (28 195 197)
      (29 420 421)
      (30 40 50)
      (30 72 78)
      (30 224 226)
      (31 480 481)
      (32 60 68)
      (32 126 130)
      (32 255 257)
      (33 44 55)
      (33 56 65)
      (33 180 183)
      (33 544 545)
      (34 288 290)
      (35 84 91)
      (35 120 125)
      (35 612 613)
      (36 48 60)
      (36 77 85)
      (36 105 111)
      (36 160 164)
      (36 323 325)
      (37 684 685)
      (38 360 362)
      (39 52 65)
      (39 80 89)
      (39 252 255)
      (39 760 761)
      (40 42 58)
      (40 75 85)
      (40 96 104)
      (40 198 202)
      (40 399 401)
      (41 840 841)
      (42 56 70)
      (42 144 150)
      (42 440 442)
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      (44 483 485)
      (45 60 75)
      (45 108 117)
      (45 200 205)
      (45 336 339)
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      (50 624 626)
      (51 68 85)
      (51 140 149)
      (51 432 435)
      (52 165 173)
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      (54 728 730)
      (55 132 143)
      (55 300 305)
      (56 90 106)
      (56 105 119)
      (56 192 200)
      (56 390 394)
      (56 783 785)
      (57 76 95)
      (57 176 185)
      (57 540 543)
      (58 840 842)
      (60 63 87)
      (60 80 100)
      (60 91 109)
      (60 144 156)
      (60 175 185)
      (60 221 229)
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      (60 448 452)
      (60 899 901)
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      (80 315 325)
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      (84 187 205)
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      (96 247 265)
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      (96 765 771)
      (98 336 350)
      (99 132 165)
      (99 168 195)
      (99 440 451)
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      (564 752 940)
      (567 756 945)
      (570 760 950)
      (573 764 955)
      (576 660 876)
      (576 768 960)
      (576 943 1105)
      (579 772 965)
      (580 609 841)
      (580 741 941)
      (582 776 970)
      (585 648 873)
      (585 780 975)
      (585 928 1097)
      (588 784 980)
      (588 945 1113)
      (591 788 985)
      (594 608 850)
      (594 792 990)
      (595 600 845)
      (595 924 1099)
      (597 796 995)
      (600 630 870)
      (600 800 1000)
      (600 910 1090)
      (603 804 1005)
      (606 808 1010)
      (609 812 1015)
      (612 759 975)
      (612 816 1020)
      (615 728 953)
      (615 820 1025)
      (616 663 905)
      (616 735 959)
      (616 870 1066)
      (616 990 1166)
      (618 824 1030)
      (620 651 899)
      (620 861 1061)
      (621 672 915)
      (621 828 1035)
      (624 715 949)
      (624 832 1040)
      (624 918 1110)
      (627 836 1045)
      (630 840 1050)
      (633 844 1055)
      (636 848 1060)
      (638 720 962)
      (639 852 1065)
      (640 672 928)
      (640 924 1124)
      (642 856 1070)
      (644 960 1156)
      (645 812 1037)
      (645 860 1075)
      (648 864 1080)
      (650 720 970)
      (651 868 1085)
      (654 872 1090)
      (657 876 1095)
      (660 693 957)
      (660 779 1021)
      (660 880 1100)
      (660 989 1189)
      (663 884 1105)
      (665 780 1025)
      (666 888 1110)
      (669 892 1115)
      (672 754 1010)
      (672 770 1022)
      (672 896 1120)
      (675 816 1059)
      (675 900 1125)
      (678 904 1130)
      (680 714 986)
      (681 908 1135)
      (682 840 1082)
      (684 912 1140)
      (684 975 1191)
      (687 916 1145)
      (690 920 1150)
      (693 924 1155)
      (696 697 985)
      (696 928 1160)
      (699 932 1165)
      (700 735 1015)
      (700 855 1105)
      (702 936 1170)
      (704 840 1096)
      (704 903 1145)
      (705 940 1175)
      (705 992 1217)
      (708 944 1180)
      (711 948 1185)
      (714 720 1014)
      (714 952 1190)
      (715 792 1067)
      (717 956 1195)
      (720 756 1044)
      (720 825 1095)
      (720 960 1200)
      (723 964 1205)
      (726 968 1210)
      (729 972 1215)
      (731 780 1069)
      (732 976 1220)
      (735 980 1225)
      (736 930 1186)
      (738 984 1230)
      (740 777 1073)
      (741 988 1235)
      (744 817 1105)
      (744 992 1240)
      (747 996 1245)
      (750 1000 1250)
      (756 825 1119)
      (760 798 1102)
      (765 868 1157)
      (768 880 1168)
      (780 819 1131)
      (780 864 1164)
      (792 806 1130)
      (792 945 1233)
      (798 936 1230)
      (799 960 1249)
      (800 840 1160)
      (816 935 1241)
      (820 861 1189)
      (828 896 1220)
      (832 855 1193)
      (833 840 1183)
      (840 882 1218)
      (845 936 1261)
      (858 920 1258)
      (860 903 1247)
      (864 990 1314)
      (880 924 1276)
      (884 987 1325)
      (891 912 1275)
      (893 924 1285)
      (900 945 1305)
      (920 966 1334)
      (940 987 1363)
      (952 960 1352))
    > (length list-list)
    1034
    >
    
    "You can't cheat an honest man. Never give a sucker an even break, or smarten up a chump." - W.C.Fields

  2. #2
    Hi Dan,

    About 5 weeks ago I thought about this problem in 3D. This is what I came up with.

    I am interested in something as close to a cube as possible, so I restricted the range.

    Among the more interesting results were: sides 42:42:49--->diagonal 77

    simplifying to 6:6:7--->11

    The simplest is 1:2:2--->3


      uses "oxygen"
    
      dim as string src="
    
      '=============================
      ' the integer diagonal problem
      '=============================
      '
      'looking for diagonals which have a simple ratio to the lengths of the sides.
    
      ' diagonal=sqrt x*x + y*y + z*z
    
      ' examples:
      '========================
      '
      ' 2D 3 4 --> diagonal 5
      ' 3D 2 3 6 --> diagonal 7
    
      '========================
    
      '==============================
      '3D TABLE: DIAGONAL OF A CUBOID
      '==============================
    
    
      %      max  100 'max diagonal
      sys    b,c,e,i,j,k,l,d[max]
      single m
      string pr,cr,se
    
      cr=chr(13)+chr(10)
      se=chr(9) 'tab
      '
      '===============================
      e=100       'max diagonal
      b=2         'min start
      m=.4        'min side/diag ratio
      '===============================
      '
      pr="TABLE OF CUBOID DIAGONAL INTEGERS" cr cr
      pr+="n" se "x" se "y" se "z" se "diagonal" cr
    
      for i=1 to e
        d[i]=i*i
      next
    
      do
        for i=b to e
          for j= i to e
            for k=j to e
              for l=k to e
                if d[i]+d[j]+d[k]=d[l] and i>e*m then
                  c+=1
                  if c>40 then exit do 'PROCESS FIRST 40 VALUES ONLY
                  pr+="" c se i se j se k se l cr
                end if
              next
            next
          next
        next
        exit do
      end do
    
      print pr
    
      "
    
      o2_asmo src
      if len(o2_error) then
        msgbox 0,o2_error
        stop
      else
        o2_exec
      end if
    
    Last edited by Charles Pegge; 25-05-2011 at 11:13. Reason: tweaks and new discoveries

  3. #3
    thinBasic MVPs danbaron's Avatar
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    Hi Charles.

    It's interesting. I think I see what you mean, higher dimensional analogs of 2D shapes, having all integer dimensions.

    I agree with your results.

    I'll think about it too.

    Dan

    "You can't cheat an honest man. Never give a sucker an even break, or smarten up a chump." - W.C.Fields

  4. #4
    Member REDEBOLT's Avatar
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    Question How can I copy code from a listing?

    If I do a ctrl-c, it selects the entire post.
    If try to drag the cursor, I only get part of the listing.
    I have selected "Reply with Quote" and then select whats inside the code tags.
    This works, but when I paste into Thinair, the listing is double-spaced.

    Anyone have a better method?



    Regards,
    Bob

  5. #5
    That's a good idea Bob. The double spacing may be a browser problem. It does not happen with Google Chrome

    The best solution is for us to post attachments if the programs are of any significant length.

    Charles
    Attached Files Attached Files

  6. #6
    thinBasic MVPs danbaron's Avatar
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    I think for a 3D right triangle, the Pythagorean theorem would become the same as your equation,

    a^2 + b^2 + c^2 = d^2.

    Here are the results for a, b, c and d all integers; and, a, b and c, not exceeding 100.

    There were 171,700 trials, and, 890 successes.



    ' code --------------------------------------------------------------------------------------------------------------------------------------
    
    #lang racket
    
    (define list-list (list))
    
    (define (list-3D-integer-right-triangles n)
      (define a 0)
      (define b 0)
      (define c 0)
      (define d 0)
      (define count1 0)
      (define count2 0)
      (define count3 0)
      (define tot-count 0)
      (set! list-list null)
      (define (loop1)
        (set! count1 (add1 count1))
        (cond
          ((> count1 n) (values tot-count (length list-list) list-list))
          (else
           (set! count2 (- count1 1))
           (loop2)
           (loop1))))
      (define (loop2)
        (set! count2 (add1 count2))
        (cond
          ((> count2 n))
          (else       
           (set! count3 (- count2 1))
           (loop3)
           (loop2))))
      (define (loop3)
        (set! count3 (add1 count3))
        (cond
          ((> count3 n))
          (else
           (set! tot-count (add1 tot-count))
           (set! a count1)
           (set! b count2)
           (set! c count3)
           (set! d (sqrt (+ (expt a 2) (expt b 2) (expt c 2))))
           (when (= d (floor d)) (set! list-list (append list-list (list (list a b c d)))))
           (loop3))))
      (loop1))
    
    ' REPL interactions -------------------------------------------------------------------------------------------------------------------------
    
    Welcome to DrRacket, version 5.1 [3m].
    Language: racket.
    > (list-3D-integer-right-triangles 100)
    171700
    890
    '((1 2 2 3)
      (1 4 8 9)
      (1 6 18 19)
      (1 8 32 33)
      (1 10 50 51)
      (1 12 12 17)
      (1 12 72 73)
      (1 14 98 99)
      (1 18 30 35)
      (1 22 46 51)
      (1 28 76 81)
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      (1 34 38 51)
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      (1 68 80 105)
      (1 70 70 99)
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      (2 4 4 6)
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      (2 7 26 27)
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    >
    
    Last edited by danbaron; 26-05-2011 at 07:43.
    "You can't cheat an honest man. Never give a sucker an even break, or smarten up a chump." - W.C.Fields

  7. #7
    thinBasic MVPs danbaron's Avatar
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    You can see below (I didn't show the lists), that if you allow the maximum dimension for one of the three "sides" of the triangle to be 700, then, there are 57,411,900 trials, and, 41,992 successes.

    Incidentally, you said you wanted to get as close to a cube as you could.

    I think it is impossible to get a cube if a, b, c and d are all integers.

    If, a = b = c, then, you basically have,

    3 * x^2 = y^2.

    In that case,

    y = sqrt(3) * x.

    ' REPL interactions ----------------------------------------------------------------------------------------------------
    
    Welcome to DrRacket, version 5.1 [3m].
    Language: racket.
    > (list-3D-integer-right-triangles 400)
    10746800
    13801
    > (list-3D-integer-right-triangles 500)
    20958500
    21436
    > (list-3D-integer-right-triangles 600)
    36180200
    30932
    > (list-3D-integer-right-triangles 700)
    57411900
    41992
    >
    

    "You can't cheat an honest man. Never give a sucker an even break, or smarten up a chump." - W.C.Fields

  8. #8
    Dan,

    It should get easier as we go up the dimensions as there are more degrees of freedom.

    the 4d hypoteneuse has a very simple solution:

    1:1:1:1--->2.

    Similarly 9d: 1:1:1:1:1:1:1:1:1--->3

    and 16d, 25d, 36d likewise.


    Charles

  9. #9
    thinBasic MVPs danbaron's Avatar
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    Yes.

    (I was not going to put this sentence, but, the forum complained that my message must be at least 10 characters.)



    Dan

    "You can't cheat an honest man. Never give a sucker an even break, or smarten up a chump." - W.C.Fields

  10. #10
    Yes .

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