* Suppose there is a stick the length of which is an integer as indicated in column 1, say 10. The question is to cut this stick into 3 pieces each of which is an integer length. In how many ways, the cutting can be done? Either, the cutting will  be such that each stick is of different length ( as indicated in column 3 ; in this case there are 4 ways to do so) or at least 2 pieces will have same length (as indicated in column 5; in this case , there are 4 ways to do so). If stick length is given (as in column 1), task is how to calculate the no. of ways 3 pieces can be made so that each piece is an integer? The data below have been compiled with actual experiment to understand the pattern and predict results for any value of stick length. To get an answer, simply put the stick length in integer positive value and submit. SUM of 3 numbers -N(1) ol(2) no. of combination--without repetition (ca) (3) or(4) no. of combinations --with repetition (cb) (5) total no. of combinations (c) (6) join (7) oc (8) Stick Length in Integer If a: b:c:, then a':b':c':a/a':b/b':c/c':(a/a')+(b/b')+(c/c'): abc:a'b'c':(abc/a'b'c'):len1=(a+b+c):(len1*a'b'c'):√(len1*a'b'c'): Join ** The series of oc term is 3,4,6,7,8,10,11,12,...... If oc term is 2, the series has 3,4. It has to be recast to a series of numbers in AP keeping the first term which is 3 and common difference which is 1. If oc term is 2, recast term is also 2 since 3,4 are already in AP. Total no. of arrangement * here help of http://www.mathaddict.net/series15a.htm has been taken. Arrangement with repetition * To find the no. of arrangements with repetition, if stick length is even, it is (length-2)/2 ; or else it is (length-1)/2. Arrangement without repetition (a/a')+(b/b')+(c/c') has minimum value of 3. No. of equilateral triangles that can be formed with 3 numbers whose sum is N (A) No. of isosceles triangles that can be formed with 3 numbers whose sum is N (B) No. of triangles with similar sides No. of triangles with dissimilar sides that can be formed with 3 numbers whose sum is N No. of child arrangement with repetition No. of non-child with repetition No. of child arrangement without repetition from    or No. of non-child without repetition

TABLE-2

 SUM of 3 numbers (1) no. of combination--without repetition ; triplets- [a,b,c] no. of combinations --with repetition (cb)lets-[a,b,c] 03 0 1(1,1,1) 04 0 1(1,1,2) 05 0 2(1,1,3)(2,2,1) 06 1(1,2,3) 2(2,2,2)(1,1,4) 07 1(1,2,4) 3(1,1,5)(2,2,3)(3,3,1) 08 2(1,2,5)(1,3,4) 3(1,1,6)(2,2,4)(3,3,2) 09 3(1,2,6)(1,3,5)(2,3,4)(24) 4(1,1,7)(2,2,5)(3,3,3)(4,4,1) 10 4(1,2,7)(1,3,6)(1,4,5)(2,3,5) 4(1,1,8)(2,2,6)(3,3,4)(4,4,2) 11 5(1,2,8)(1,3,7)(1,4,6)(2,3,6)(2,4,5)(40) 5(1,1,9)(2,2,7)(3,3,5)(4,4,3)(5,5,1) 12 7(1,2,9)(1,3,8)(1,4,7)(1,5,6)(2,3,7)(2,4,6)(3,4,5)(60) 5(1,1,10)(2,2,8)(3,3,6)(4,4,4)(5,5,2) 13 8(1,2,10)(1,3,9)(1,4,8)(1,5,7)(2,3,8)(2,4,7)(2,5,6)(60)(3,4,6)(72) 6(1,1,11)(2,2,9)(3,3,7)(4,4,5)(5,5,3)(6,6,1) 14 10(1,2,11)(1,3,10)(1,4,9)(1,5,8)(1,6,7)(2,3,9)(2,4,8)(2,5,7)(3,4,7)(3,5,6)(90) 6(1,1,12)(2,2,10)(3,3,8)(4,4,6)(5,5,4)(6,6,2) 15 12 (1,2,12)(1,3,11)(1,4,10)(1,5,9)(1,6,8)(2,3,10)(2,4,9)(2,5,8)(2,6,7)(84)(3,4,8)(3,5,7)(105)(4,5,6)(120) 7(1,1,13)(2,2,11)(3,3,9)(4,4,7)(5,5,5)(6,6,3)(7,7,1) 16 14 ((1,2,13)(1,3,12)(1,4,11)(1,5,10)(1,6,9)(1,7,8)(2,3,11)(2,4,10)(2,5,9)(2,6,8)(3,4,9)(3,5,8)(3,6,7)(126)(4,5,7)(140) 7(1,1,14)(2,2,12)(3,3,10)(4,4,8)(5,5,6)(6,6,4)(7,7,2) 17 16(1,2,14)(1,3,13)(1,4,12)(1,5,11)(1,6,10)(1,7,9)(2,3,12)(2,4,11)(2,5,10)(2,6,9)(2,7,8)(112)(3,4,10)(3,5,9)(3,6,8)(144)(4,5,8)(160)(4,6,7)(168) 8(1,1,15)(2,2,13)(3,3,11)(4,4,9)(5,5,7)(6,6,5)(7,7,3)(8,8,1) 18 19(1,2,15)(1,3,14)(1,4,13)(1,5,12)(1,6,11)(1,7,10)(1,8,9)(2,3,13)(2,4,12)(2,5,11)(2,6,10)(2,7,9)(3,4,11)(3,5,10)(3,6,9)(3,7,8)(168)(4,5,9)(4,6,8)(192)(5,6,7)(210) 8(1,1,16)(2,2,14)(3,3,12)(4,4,10)(5,5,8)(6,6,6)(7,7,4)(8,8,2) 19 21(1,2,16)(1,3,15)(1,4,14)(1,5,13)(1,6,12)(1,7,11)(1,8,10)(2,3,14)(2,4,13)(2,5,12)(2,6,11)(2,7,10)(2,8,9)(144)(3,4,12)(3,5,11)(3,6,10)(3,7,9)(189)(4,5,10)(4,6,9)(216)(4,7,8)(224)(5,6,8)(240) 9(1,1,17)(2,2,15)(3,3,13)(4,4,11)(5,5,9)(6,6,7)(7,7,5)(8,8,3)(9,9,1) 20 24(1,2,17)(1,3,16)(1,4,15)(1,5,14)(1,6,13)(1,7,12)(1,8,11)(1,9,10)(2,3,15)(2,4,14)(2,5,13)(2,6,12)(2,7,11)(2,8,10)(3,4,13)(3,5,12)(3,6,11)(3,7,10)(3,8,9)(216)(4,5,11)(4,6,10)(4,7,9)(252)(5,6,9)(270)(5,7,8)(280) 9(1,1,18)(2,2,16)(3,3,14)(4,4,12)(5,5,10)(6,6,8)(7,7,6)(8,8,4)(9,9,2) 21 27(1,2,18)(1,3,17)(1,4,16)(1,5,15)(1,6,14)(1,7,13)(1,8,12)(1,9,11)(2,3,16)(2,4,15)(2,5,14)(2,6,13)(2,7,12)(2,8,11)(2,9,10)(180)(3,4,14)(3,5,13)(3,6,12)(3,7,11)(3,8,10)(240)(4,5,12)(4,6,11)(4,7,10)(280)(4,8,9)(288)(5,6,10)(300)(5,7,9)(315)(6,7,8)(336) 10(1,1,19)(2,2,17)(3,3,15)(4,4,13)(5,5,11)(6,6,9)(7,7,7)(8,8,5)(9,9,3)(10,10,1) 22 30(1,2,19)(1,3,18)(1,4,17)(1,5,16)(1,6,15)(1,7,14)(1,8,13)(1,9,12)(1,10,11)(2,3,17)(2,4,16)(2,5,15)(2,6,14)(2,7,13)(2,8,12)(2,9,11)(3,4,15)(3,5,14)(3,6,13)(3,7,12)(3,8,11)(3,9,10)(270)(4,5,13)(4,6,12)(4,7,11)(4,8,10)(320)(5,6,11)(5,7,10)(350)(5,8,9)(360)(6,7,9)(378) 10(1,1,20)(2,2,18)(3,3,16)(4,4,14)(5,5,12)(6,6,10)(7,7,8)(8,8,6)(9,9,4)(10,10,2) 23 33(1,2,20)(1,3,19)(1,4,18)(1,5,17)(1,6,16)(1,7,15)(1,8,14)(1,9,13)(1,10,12)(2,3,18)(2,4,17)(2,5,16)(2,6,15)(2,7,14)(2,8,13)(2,9,12)(2,10,11)(220)(3,4,16)(3,5,15)(3,6,14)(3,7,13)(3,8,12)(3,9,11)(297)(4,5,14)(4,6,13)(4,7,12)(4,8,11)(352)(4,9,10)(360)(5,6,12)(5,7,11)(385)(5,8,10)(400)(6,7,10)(420)(6,8,9)(432) 11(1,1,21)(2,2,19)(3,3,17)(4,4,15)(5,5,13)(6,6,11)(7,7,9)(8,8,7)(9,9,5)(10,10,3)(11,11,1) 24 37(1,2,21)(1,3,20)(1,4,19)(1,5,18)(1,6,17)(1,7,16)(1,8,15)(1,9,14)(1,10,13)(1,11,12)(2,3,19)(2,4,18)(2,5,17)(2,6,16)(2,7,15)(2,8,14)(2,9,13)(2,10,12)(3,4,17)(3,5,16)(3,6,15)(3,7,14)(3,8,13)(3,9,12)(3,10,11)(330)(4,5,15)(4,6,14)(4,7,13)(4,8,12)(4,9,11)(396)(5,6,13)(5,7,12)(5,8,11)(440)(5,9,10)(450)(6,7,11)(462)(6,8,10)(480)(7,8,9)(504) 11(1,1,22)(2,2,20)(3,3,18)(4,4,16)(5,5,14)(6,6,12)(7,7,10)(8,8,8)(9,9,6)(10,10,4)(11,11,2) 25 40(1,2,22)(1,3,21)(1,4,20)(1,5,19)(1,6,18)(1,7,17)(1,8,16)(1,9,15)(1,10,14)(1,11,13)(2,3,20)(2,4,19)(2,5,18)(2,6,17)(2,7,16)(2,8,15)(2,9,14)(2,10,13)(2,11,12)(264)(3,4,18)(3,5,17)(3,6,16)(3,7,15)(3,8,14)(3,9,13)(3,10,12)(360)(4,5,16)(4,6,15)(4,7,14)(4,8,13)(4,9,12)(432)(4,10,11)(440)(5,6,14)(5,7,13)(5,8,12)(480)(5,9,11)(495)(6,7,12)(504)(6,8,11)(528)(6,9,10)(540)(7,8,10)(560) 12(1,1,23)(2,2,21)(3,3,19)(4,4,17)(5,5,15)(6,6,13)(7,7,11)(8,8,9)(9,9,7)(10,10,5)(11,11,3)(12,12,1) 26 44(1,2,23)(1,3,22)(1,4,21)(1,5,20)(1,6,19)(1,7,18)(1,8,17)(1,9,16)(1,10,15)(1,11,14)(1,12,13)(2,3,21)(2,4,20)(2,5,19)(2,6,18)(2,7,17)(2,8,16)(2,9,15)(2,10,14)(2,11,13)(3,4,19)(3,5,18)(3,6,17)(3,7,16)(3,8,15)(3,9,14)(3,10,13)(3,11,12)(396)(4,5,17)(4,6,16)(4,7,15)(4,8,14)(4,9,13)(4,10,12)(480)(5,6,15)(5,7,14)(5,8,13)(5,9,12)(540)(5,10,11)(550)(6,7,13)(6,8,12)(576)(6,9,11)(594)(7,8,11)(616)(7,9,10)(630) 12(1,1,24)(2,2,22)(3,3,20)(4,4,18)(5,5,16)(6,6,14)(7,7,12)(8,8,10)(9,9,8)(10,10,6)(11,11,4)(12,12,2) 27 48(1,2,24)(1,3,23)(1,4,22)(1,5,21)(1,6,20)(1,7,19)(1,8,18)(1,9,17)(1,10,16)(1,11,15)(1,12,14)(2,3,22)(2,4,21)(2,5,20)(2,6,19)(2,7,18)(2,8,17)(2,9,16)(2,10,15)(2,11,14)(2,12,13)(312)(3,4,20)(3,5,19)(3,6,18)(3,7,17)(3,8,16)(3,9,15)(3,10,14)(3,11,13)(429)(4,5,18)(4,6,17)(4,7,16)(4,8,15)(4,9,14)(4,10,13)(520)(4,11,12)(528)(5,6,16)(5,7,15)(5,8,14)(5,9,13)(585)(5,10,12)(600)(6,7,14)(6,8,13)(624)(6,9,12)(648)(6,10,11)(660)(7,8,12)(672)(7,9,11)(693)(8,9,10)(720) 13(1,1,25)(2,2,23)(3,3,21)(4,4,19)(5,5,17)(6,6,15)(7,7,13)(8,8,11)(9,9,9)(10,10,7)(11,11,5)(12,12,3)(13,13,1) 28 52(1,2,25)(1,3,24)(1,4,23)(1,5,22)(1,6,21)(1,7,20)(1,8,19)(1,9,18)(1,10,17)(1,11,16)(1,12,15)(1,13,14)(2,3,23)(2,4,22)(2,5,21)(2,6,20)(2,7,19)(2,8,18)(2,9,17)(2,10,16)(2,11,15)(2,12,14)(3,4,21)(3,5,20)(3,6,19)(3,7,18)(3,8,17)(3,9,16)(3,10,15)(3,11,14)(3,12,13)(468)(4,5,19)(4,6,18)(4,7,17)(4,8,16)(4,9,15)(4,10,14)(4,11,13)(572)(5,6,17)(5,7,16)(5,8,15)(5,9,14)(5,10,13)(650)(5,11,12)(660)(6,7,15)(6,8,14)(6,9,13)(702)(6,10,12)(720)(7,8,13)(728)(7,9,12)(756)(7,10,11)(770)(8,9,11)(792) 13(1,1,26)(2,2,24)(3,3,22)(4,4,20)(5,5,18)(6,6,16)(7,7,14)(8,8,12)(9,9,10)(10,10,8)(11,11,6)(12,12,4)(13,13,2) 29 56(1,2,26)(1,3,25)(1,4,24)(1,5,23)(1,6,22)(1,7,21)(1,8,20)(1,9,19)(1,10,18)(1,11,17)(1,12,16)(1,13,15)(2,3,24)(2,4,23)(2,5,22)(2,6,21)(2,7,20)(2,8,19)(2,9,18)(2,10,17)(2,11,16)(2,12,15)(2,13,14)(364)(3,4,22)(3,5,21)(3,6,20)(3,7,19)(3,8,18)(3,9,17)(3,10,16)(3,11,15)(3,12,14)(504)(4,5,20)(4,6,19)(4,7,18)(4,8,17)(4,9,16)(4,10,15)(4,11,14)(616)(4,12,13)(624)(5,6,18)(5,7,17)(5,8,16)(5,9,15)(5,10,14)(700)(5,11,13)(715)(6,7,16)(6,8,15)(6,9,14)(756)(6,10,13)(780)(6,11,12)(792)(7,8,14)(784)(7,9,13)(819)(7,10,12)(840)(8,9,12)(864)(8,10,11)(880) 14(1,1,27)(2,2,25)(3,3,23)(4,4,21)(5,5,19)(6,6,17)(7,7,15)(8,8,13)(9,9,11)(10,10,9)(11,11,7)(12,12,5)(13,13,3)(14,14,1) 30 61(1,2,27)(1,3,16)(1,4,25)(1,5,24)(1,6,23)(1,7,22)(1,8,21)(1,9,20)(1,10,19)(1,11,18)(1,12,17)(1,13,16)(1,14,15)(2,3,25)(2,4,24)(2,5,23)(2,6,22)(2,7,21)(2,8,20)(2,9.19)(2,10,18)(2,11,17)(2,12,16)(2,13,15)(3,4,23)(3,5,22)(3,6,21)(3,7,20)(3,8,19)(3,9,18)(3,10,17)(3,11,16)(3,12,15)(3,13,14)(4,5,21)(4,6,20)(4,7,19)(4,8,18)(4,9,17)(4,10,16)(4,11,15)(4,12,14)(5,6,19)(5,7,18)(5,8,17)(5,9,16)(5,10,15)(5,11,14)(5,12,13)(6,7,17)(6,8,16)(6,9,15)(6,10,14)(6,11,13)(7,8,15)(7,9,14)(7,10,13)(7,11,12)(8,9,13)(8,10,12)(9,10,11) 14(1,1,28)(2,2,26)(3,3,24)(4,4,22)(5,5,20)(6,6,18)(7,7,16)(8,8,14)(9,9,12)(10,10,10)(11,11,8)(12,12,6)(13,13,4)(14,14,2) 31 65(1,2,28)(1,3,27)(1,4,26)(1,5,25)(1,6,24)(1,7,23)(1,8,22)(1,9,21)(1,10,20)(1,11,19)(1,12,18)(1,13,17)(1,14,16)(2,3,26)(2,4,25)(2,5,24)(2,6,23)(2,7,22)(2,8,21)(2,9,20)(2,10,19)(2,11,18)(2,12,17)(2,13,16)(2,14,15)(3,4,24)(3,5,23)(3,6,22)(3,7,21)(3,8,20)(3,9,19)(3,10,18)(3,11,17)(3,12,16)(3,13,15)(4,5,22)(4,6,21)(4,7,20)(4,8,19)(4,9,18)(4,10,17)(4,11,16)(4,12,15)(4,13,14)(5,6,20)(5,7,19)(5,8,18)(5,9,17)(5,10,16)(5,11,15)(5,12,14)(6,7,18)(6,8,17)(6,9,16)(6,10,15)(6,11,14)(6,12,13)(7,8,16)(7,9,15)(7,10,14)(7,11,13)(8,9,14)(8,10,13)(8,11,12)(9,10,12) 15(1,1,29)(2,2,27)(3,3,25)(4,4,23)(5,5,21)(6,6,19)(7,7,17)(8,8,15)(9,9,13)(10,10,11)(11,11,9)(12,12,7)(13,13,5)(14,14,3)(15,15,1) 32 70(1,2,29)(1,3,28)(1,4,27)(1,5,26)(1,6,25)(1,7,24)(1,8,23)(1,9,22)(1,10,21)(1,11,20)(1,12,19)(1,13,18)(1,14,17)(1,15,16)(2,3,27)(2,4,26)(2,5,25)(2,6,24)(2,7,23)(2,8,22)(2,9,21)(2,10,20)(2,11,19)(2,12,18)(2,13,17)(2,14,16)(3,4,25)(3,5,24)(3,6,23)(3,7,22)(3,8,21)(3,9,20)(3,10,19)(3,11,18)(3,12,17)(3,13,16)(3,14,15)(4,5,23)(4,6,22)(4,7,21)(4,8,20)(4,9,19)(4,10,18)(4,11,17)(4,12,16)(4,13,15)(5,6,21)(5,7,20)(5,8,19)(5,9,18)(5,10,17)(5,11,16)(5,12,15)(5,13,14)(6,7,19)(6,8,18)(6,9,17)(6,10,16)(6,11,15)(6,12,14)(7,8,17)(7,9,16)(7,10,15)(7,11,14)(7,12,13)(8,9,15)(8,10,14)(8,11,13)(9,10,13)(9,11,12) 15(1,1,30)(2,2,28)(3,3,26)(4,4,24)(5,5,22)(6,6,20)(7,7,18)(8,8,16)(9,9,14)(10,10,12)(11,11,10)(12,12,8)(13,13,6)(14,14,4)(15,15,2) 33 75(1,2,30)(1,3,29)(1,4,28)(1,5,27)(1,6,26)(1,7,25)(1,8,24)(1,9,23)(1,10,22)(1,11,21)(1,12,20)(1,13,19)(1,14,18)(1,15,17)(2,3,28)(2,4,27)(2,5,26)(2,6,25)(2,7,24)(2,8,23)(2,9,22)(2,10,21)(2,11,20)(2,12,19)(2,13,18)(2,14,17)(2,15,16)(3,4,26)(3,5,25)(3,6,24)(3,7,23)(3,8,22)(3,9,21)(3,10,20)(3,11,19)(3,12,18)(3,13,17)(3,14,16)(4,5,24)(4,6,23)(4,7,22)(4,8,21)(4,9,20)(4,10,19)(4,11,18)(4,12,17)(4,13,16)(4,14,15)(5,6,22)(5,7,21)(5,8,20)(5,9,19)(5,10,18)(5,11,17)(5,12,16)(5,13,15)(6,7,20)(6,8,19)(6,9,18)(6,10,17)(6,11,16)(6,12,15)(6,13,14)(7,8,18)(7,9,17)(7,10,16)(7,11,15)(7,12,14)(8,9,16)(8,10,15)(8,11,14)(8,12,13)(9,10,14)(9,11,13)(10,11,12) 16(1,1,31)(2,2,29)(3,3,27)(4,4,25)(5,5,23)(6,6,21)(7,7,19)(8,8,17)(9,9,15)(10,10,13)(11,11,11)(12,12,9)(13,13,7)(14,14,5)(15,15,3)(16,16,1) 34 80(1,2,31)(1,3,30)(1,4,29)(1,5,28)(1,6,27)(1,7,26)(1,8,25)(1,9,24)(1,10,23)(1,11,22)(1,12,21)(1,13,20)(1,14,19)(1,15,18)(1,16,17)(2,3,29)(2,4,28)(2,5,27)(2,6,26)(2,7,25)(2,8,24)(2,9,23)(2,10,22)(2,11,21)(2,12,20)(2,13,19)(2,14,18)(2,15,17)(3,4,27)(3,5,26)(3,6,25)(3,7,24)(3,8,23)(3,9,22)(3,10,21)(3,11,20)(3,12,19)(3,13,18)(3,14,17)(3,15,16)(4,5,25)(4,6,24)(4,7,23)(4,8,22)(4,9,21)(4,10,20)(4,11,19)(4,12,18)(4,13,17)(4,14,16)(5,6,23)(5,7,22)(5,8,21)(5,9,20)(5,10,19)(5,11,18)(5,12,17)(5,13,16)(5,14,15)(6,7,21)(6,8,20)(6,9,19)(6,10,18)(6,11,17)(6,12,16)(6,13,15)(7,8,19)(7,9,18)(7,10,17)(7,11,16)(7,12,15)(7,13,14)(8,9,17)(8,10,16)(8,11,15)(8,12,14)(9,10,15)(9,11,14)(9,12,13)(10,11,13) 16(1,1,32)(2,2,30)(3,3,28)(4,4,26)(5,5,24)(6,6,22)(7,7,20)(8,8,18)(9,9,16)(10,10,14)(11,11,12)(12,12,10)(13,13,8)(14,14,6)(15,15,4)(16,16,2) 35 85(1,2,32)(1,3,31)(1,4,30)(1,5,29)(1,6,28)(1,7,27)(1,8,26)(1,9,25)(1,10,24)(1,11,23)(1,12,22)(1,13,21)(1,14,20)(1,15,19)(1,16,18)(2,3,30)(2,4,29)(2,5,28)(2,6,27)(2,7,26)(2,8,25)(2,9,24)(2,10,23)(2,11,22)(2,12,21)(2,13,20)(2,14,19)(2,15,18)(2,16,17)(3,4,28)(3,5,27)(3,6,26)(3,7,25)(3,8,24)(3,9,23)(3,10,22)(3,11,21)(3,12,20)(3,13,19)(3,14,18)(3,15,17)(4,5,26)(4,6,25)(4,7,24)(4,8,23)(4,9,22)(4,10,21)(4,11,20)(4,12,19)(4,13,18)(4,14,17)(4,15,16)(5,6,24)(5,7,23)(5,8,22)(5,9,21)(5,10,20)(5,11,19)(5,12,18)(5,13,17)(5,14,16)(6,7,22)(6,8,21)(6,9,20)(6,10,19)(6,11,18)(6,12,17)(6,13,16)(6,14,15)(7,8,20)(7,9,19)(7,10,18)(7,11,17)(7,12,16)(7,13,15)(8,9,18)(8,10,17)(8,11,16)(8,12,15)(8,13,14)(9,10,16)(9,11,15)(9,12,14)(10,11,14)(10,12,13) 36 91(1,2,33)(1,3,32)(1,4,31)(1,5,30)(1,6,29)(1,7,28)(1,8,27)(1,9,26)(1,10,25)(1,11,24)(1,12,23)(1,13,22)(1,14,21)(1,15,20)(1,16,19)(1,17,18)(2,3,31)(2,4,30)(2,5,29)(2,6,28)(2,7,27)(2,8,26)(2,9,25)(2,10,24)(2,11,23)(2,12,22)(2,13,21)(2,14,20)(2,15,19)(2,16,18)(3,4,29)(3,5,28)(3,6,27)(3,7,26)(3,8,25)(3,9,24)(3,10,23)(3,11,22)(3,12,21)(3,13,20)(3,14,19)(3,15,18)(3,16,17)(4,5,27)(4,6,26)(4,7,25)(4,8,24)(4,9,23)(4,10,22)(4,11,21)(4,12,20)(4,13,19)(4,14,18)(4,15,17)(5,6,25)(5,7,24)(5,8,23)(5,9,22)(5,10,21)(5,11,20)(5,12,19)(5,13,18)(5,14,17)(5,15,16)(6,7,23)(6,8,22)(6,9,21)(6,10,20)(6,11,19)(6,12,18)(6,13,17)(6,14,16)(7,8,21)(7,9,20)(7,10,19)(7,11,18)(7,12,17)(7,13,16)(7,14,15)(8,9,19)(8,10,18)(8,11,17)(8,12,16)(8,13,15)(9,10,17)(9,11,16)(9,12,15)(9,13,14)(10,11,15)(10,12,14)(11,12,13) 37 96(1,2,34)(1,3,33)(1,4,32)(1,5,31)(1,6,30)(1,7,29)(1,8,28)(1,9,27)(1,10,26)(1,11,25)(1,12,24)(1,13,23)(1,14,22)(1,15,21)(1,16,20)(1,17,19)(2,3,32)(2,4,31)(2,5,30)(2,6,29)(2,7,28)(2,8,27)(2,9,26)(2,10,25)(2,11,24)(2,12,23)(2,13,22)(2,14,21)(2,15,20)(2,16,19)(2,17,18)(3,4,30)(3,5,29)(3,6,28)(3,7,27)(3,8,26)(3,9,25)(3,10,24)(3,11,23)(3,12,22)(3,13,21)(3,14,20)(3,15,19)(3,16,18)(4,5,28)(4,6,27)(4,7,26)(4,8,25)(4,9,24)(4,10,23)(4,11,22)(4,12,21)(4,13,20)(4,14,19)(4,15,18)(4,16,17)(5,6,26)(5,7,25)(5,8,24)(5,9,23)(5,10,22)(5,11,21)(5,12,20)(5,13,19)(5,14,18)(5,15,17)(6,7,24)(6,8,23)(6,9,22)(6,10,21)(6,11,20)(6,12,19)(6,13,18)(6,14,17)(6,15,16)(7,8,22)(7,9,21)(7,10,20)(7,11,19)(7,12,18)(7,13,17)(7,14,16)(8,9,20)(8,10,19)(8,11,18)(8,12,17)(8,13,16)(8,14,15)(9,10,18)(9,11,17)(9,12,16)(9,13,15)(10,11,16)(10,12,15)(10,13,14)(11,12,14) 38 102(1,2,35)(1,3,34)(1,4,33)(1,5,32)(1,6,31)(1,7,30)(1,8,29)(1,9,28)(1,10,27)(1,11,26)(1,12,25)(1,13,24)(1,14,23)(1,15,22)(1,16,21)(1,17,20)(1,18,19)(2,3,33)(2,4,32)(2,5,31)(2,6,30)(2,7,29)(2,8,28)(2,9,27)(2,10,26)(2,11,25)(2,12,24)(2,13,23)(2,14,22)(2,15,21)(2,16,20)(2,17,19)(3,4,31)(3,5,30)(3,6,29)(3,7,28)(3,8,27)(3,9,26)(3,10,25)(3,11,24)(3,12,23)(3,13,22)(3,14,21)(3,15,20)(3,16,19)(3,17,18)(4,5,29)(4,6,28)(4,7,27)(4,8,26)(4,9,25)(4,10,24)(4,11,23)(4,12,22)(4,13,21)(4,14,20)(4,15,19)(4,16,18)(5,6,27)(5,7,26)(5,8,25)(5,9,24)(5,10,23)(5,11,22)(5,12,21)(5,13,20)(5,14,19)(5,15,18)(5,16,17)(6,7,25)(6,8,24)(6,9,23)(6,10,22)(6,11,21)(6,12,20)(6,13,19)(6,14,18)(6,15,17)(7,8,23)(7,9,22)(7,10,21)(7,11,20)(7,12,19)(7,13,18)(7,14,17)(7,15,16)(8,9,21)(8,10,20)(8,11,19)(8,12,18)(8,13,17)(8,14,16)(9,10,19)(9,11,18)(9,12,17)(9,13,16)(9,14,15)(10,11,17)(10,12,16)(10,13,15)(11,12,15)(11,13,14) 39 107(1,2,36)(1,3,35)(1,4,34)(1,5,33)(1,6,32)(1,7,31)(1,8,30)(1,9,29)(1,10,28)(1,11,27)(1,12,26)(1,13,25)(1,14,24)(1,15,23)(1,16,22)(1,17,21)(1,18,20)(2,3,34)(2,4,33)(2,5,32)(2,6,31)(2,7,30)(2,8,29)(2,9,28)(2,10,27)(2,11,26)(2,12,25)(2,13,24)(2,14,23)(2,15,22)(2,16,21)(2,17,20)(2,18,19)(3,4,32)(3,5,31)(3,6,30)(3,7,29)(3,8,28)(3,9,27)(3,10,26)(3,11,25)(3,12,24)(3,13,23)(3,14,22)(3,15,21)(3,16,20)(3,17,19)(4,5,30)(4,6,29)(4,7,28)(4,8,27)(4,9,26)(4,10,25)(4,11,24)(4,12,23)(4,13,22)(4,14,21)(4,15,20)(4,16,19)(4,17,18)(5,6,28)(5,7,27)(5,8,26)(5,9,25)(5,10,24)(5,11,23)(5,12,22)(5,13,21)(5,14,20)(5,15,19)(5,16,18)(6,7,26)(6,8,25)(6,9,24)(6,10,23)(6,11,22)(6,12,21)(6,13,20)(6,14,19)(6,15,18)(6,16,17)(7,8,24)(7,9,23)(7,10,22)(7,11,21)(7,12,20)(7,13,19)(7,14,18)(7,15,17)(8,9,22)(8,10,21)(8,11,20)(8,12,19)(8,13,18)(8,14,17)(8,15,16)(9,10,20)(9,11,19)(9,12,18)(9,13,17)(9,14,16)(10,11,18)(10,12,17)(10,13,16)(10,14,15)(11,12,16)(11,13,15)(12,13,14) 40 114(1,2,37)(1,3,36)(1,4,35)(1,5,34)(1,6,33)(1,7,32)(1,8,31)(1,9,30)(1,10,29)(1,11,28)(1,12,27)(1,13,26)(1,14,25)(1,15,24)(1,16,23)(1,17,22)(1,18,21)(1,19,20)(2,3,35)(2,4,34)(2,5,33)(2,6,32)(2,7,31)(2,8,30)(2,9,29)(2,10,28)(2,11,27)(2,12,26)(2,13,25)(2,14,24)(2,15,23)(2,16,22)(2,17,21)(2,18,20)(3,4,33)(3,5,32)(3,6,31)(3,7,30)(3,8,29)(3,9,28)(3,10,27)(3,11,26)(3,12,25)(3,13,24)(3,14,23)(3,15,22)(3,16,21)(3,17,20)(3,18,19)(4,5,31)(4,6,30)(4,7,29)(4,8,28)(4,9,27)(4,10,26)(4,11,25)(4,12,24)(4,13,23)(4,14,22)(4,15,21)(4,16,20)(4,17,19)(5,6,29)(5,7,28)(5,8,27)(5,9,26)(5,10,25)(5,11,24)(5,12,23)(5,13,22)(5,14,21)(5,15,20)(5,16,19)(5,17,18)(6,7,27)(6,8,26)(6,9,25)(6,10,24)(6,11,23)(6,12,22)(6,13,21)(6,14,20)(6,15,19)(6,16,18)(7,8,25)(7,9,24)(7,10,23)(7,11,22)(7,12,21)(7,13,20)(7,14,19)(7,15,18)(7,16,17)(8,9,23)(8,10,22)(8,11,21)(8,12,20)(8,13,19)(8,14,18)(8,15,17)(9,10,21)(9,11,20)(9,12,19)(9,13,18)(9,14,17)(9,15,16)(10,11,19)(10,12,18)(10,13,17)(10,14,16)(11,12,17)(11,13,16)(11,14,15)(12,13,15) 41 120(1,2,38)(1,3,37)(1,4,36)(1,5,35)(1,6,34)(1,7,33)(1,8,32)(1,9,31)(1,10,30)(1,11,29)(1,12,28)(1,13,27)(1,14,26)(1,15,25)(1,16,24)(1,17,23)(1,18,22)(1,19,21)(2,3,36)(2,4,35)(2,5,34)(2,6,33)(2,7,32)(2,8,31)(2,9,30)(2,10,29)(2,11,28)(2,12,27)(2,13,26)(2,14,25)(2,15,24)(2,16,23)(2,17,22)(2,18,21)(2,19,20)(760)(3,4,34)(3,5,33)(3,6,32)(3,7,31)(3,8,30)(3,9,29)(3,10,28)(3,11,27)(3,12,26)(3,13,25)(3,14,24)(3,15,23)(3,16,22)(3,17,21)(3,18,20)(1080)(4,5,32)(4,6,31)(4,7,30)(4,8,29)(4,9,28)(4,10,27)(4,11,26)(4,12,25)(4,13,24)(4,14,23)(4,15,22)(4,16,21)(4,17,20)(1360)(4,18,19)(1368)(5,6,30)(5,7,29)(5,8,28)(5,9,27)(5,10,26)(5,11,25)(5,12,24)(5,13,23)(5,14,22)(5,15,21)(5,16,20)(1600)(5,17,19)(1615)(6,7,28)(6,8,27)(6,9,26)(6,10,25)(6,11,24)(6,12,23)(6,13,22)(6,14,21)(6,15,20)(1800)(6,16,19)(1824)(6,17,18)(1836)(7,8,26)(7,9,25)(7,10,24)(7,11,23)(7,12,22)(7,13,21)(7,14,20)(1960)(7,15,19)(1995)(7,16,18)(2016)(8,9,24)(8,10,23)(8,11,22)(8,12,21)(8,13,20)(2080)(8,14,19)(2128)(8,15,18)(2160)(8,16,17)(2176)(9,10,22)(9,11,21)(9,12,20)(2160)(9,13,19)(2223)(9,14,18)(2268)(9,15,17)(2295)(10,11,20)(2200)(10,12,19)(2280)(10,13,18)(2340)(10,14,17)(2380)(10,15,16)(2400)(11,12,18)(2376)(11,13,17)(2431)(11,14,16)(2464)(12,13,16)(2496)(12,14,15)(2520)

* Blue colored triplets indicate where triangles can be formed & red colored triplets indicate triangles are right angled.

* Green colored figures indicate product of the triplets where triangles are formed.

* For no. of combinations, triplets are represented as (a,b,c). For every (a,b,c)  which forms a triangle (given by blue/red color) in without repetition arrangement, there exists an (a',b',c') within the same no.  such that (a'.b',c') may or may not form triangles. And For every (a,b,c)  which forms a triangle (given by blue color)  with repetition arrangement, there exists an (a',b',c') within the same no.  such that (a'.b',c') also  forms triangles in some cases and do not form triangles in some cases. a'=b+c-a, b'=a+c-b, c'=a+b-c. We call (a'.b',c') child triplet. Exa:- For 18,if triangle triplet is (4,6,8), child triplet is (10,6,2) which lies within 18 but does not form triangle. For arrangement with repetition, if triangle triplet is (5,5,8), child triplet is (8,8,2) which lies in triangle triplets whereas if triangle triplet is (7,7,4), child triplet (4,4,10) lies out of the triangle triplets. Just put the value of (a,b,c) for triangle triplets(either from with or without repetition arrangement) and submit and the child (a',b',c') will appear. For instance in case of no. 18, put (3,7,8) and child (2,4,12) will appear.

* In the table, bracket figures are (non-child + child). - means child triplet does not exist.

* If the triplets are not pythagorean triplets, (len1*a'b'c') will not be a perfect square.

(*) To find total arrangement , visit http://www.mathaddict.net/comm4b.htm . This is an arrangement with ordering. For example, if sum of 3 numbers is 6, arrangements without ordering are (1,2,3),(2,2,2)(1,1,4). But on ordering, (1,2,3) becomes (1,2,3)(1,3,2),(2,1,3),(3,1,2),(2,3,1),(3,2,1) i.e 3! for all 3 elements being dissimilar. (2,2,2) remains same as all elements are similar. (1,1,4) becomes (1,1,4),(1,4,1),(4,1,1) i.e C(3,2)=3. Thus total arrangement becomes 10 from 3.

 SUM of 3 numbers Arrangement-without repetition(non-child +child) Arrangement where triangle formed Arrangement-with repetition(non- child + child) Arrangement where triangle formed Total arrangement(non-child+child) Total arrangement where triangle formed Total arrangementwith ordering (*) 3 0 0 1 (-+0) 1 1(0+0) 1 1 4 0 0 1 (1+-) 0 1(1+0) 0 3 5 0 0 2 (0+1) 1 2(0+1) 1 6 6 1 (1+-) 0 2 (1+0) 1 3(2+0) 1 10 7 1 (1+-) 0 3 (0+1) 2 4(1+1) 2 15 8 2 (2+-) 0 3 (1+1) 1 5(3+1) 1 21 9 3 (1+1) 1 4 (1+1) 2 7(2+2) 3 28 10 4 (4+-) 0 4 (1+1) 2 8(5+1) 2 36 11 5 (3+1) 1 5 (1+1) 3 10(4+2) 4 45 12 7 (5+1) 1 5 (2+1) 2 12(7+2) 3 55 13 8 (4+2) 2 6 (1+2) 3 14(5+4) 5 66 14 10 (8+1) 1 6 (2+1) 3 16(10+2) 4 78 15 12 (7+2) 3 7 (1+2) 4 19(8+4) 7 91 16 14 (10+2) 2 7 (2+2) 3 21(12+4) 5 17 16  (8+4) 4 8 (2+2) 4 24(10+6) 8 18 19 (14+2) 3 8 (2+2) 4 27(16+4) 7 19 21 (12+4) 5 9 (2+2) 5 30(14+6) 10 20 24 (16+4) 4 9 (3+2) 4 33(19+6) 8 21 27 (14+6) 7 10(2+3) 5 37(16+9) 12 22 30 (21+4) 5 10(3+2) 5 40(24+6) 10 23 33 (19+6) 8 11(2+3) 6 44(21+9) 14 24 37 (24+6) 7 11(3+3) 5 48(27+9) 12 25 40 (21+9) 10 12(3+3) 6 52(24+12) 16 26 44 (30+6) 8 12(3+3) 6 56(33+9) 14 27 48 (27+9) 12 13(3+3) 7 61(30+12) 19 28 52 (33+9) 10 13(4+3) 6 65(37+12) 16 29 56 (30+12) 14 14(3+4) 7 70(33+16) 21 30 61 (40+9) 12 14(4+3) 7 75(44+12) 19 31 65 (37+12) 16 15(3+4) 8 80(40+16) 24 32 70 (44+12) 14 15(4+4) 7 85(48+16) 21 33 75 (40+16) 19 16(4+4) 8 91(44+20) 27 34 80 (52+12) 16 16(4+4) 8 96(56+16) 24 35 85 (48+16) 21 17(4+4) 9 36 91 (56+16) 19 17(5+4) 8 37 96 (52+20) 24 18(4+5) 9 38 102(65+16) 21 18(5+4) 9 39 108(61+20) 27 19(4+5) 10 40 114(70+20) 24 19(5+5) 9 41 120(65+25) 30 20(5+5) 10