bài tập bài toán chiếc túi xách

TAP.CCT a. Ph5t bidu bdi to5n: Bii tQp 1: Bii toSn chi€'c triix5ch G\/ MOt c6i kho chfa n loai d6 v6t c6 kich thr/6c vd gi5 tri khiSc nhau. Cu thd: Loai db vgt i (i=1 n) c6: - Kich cd m[i] e N. - Gi5 tri c[i] e R - Sd lrtOng: khdng han ch6 M6t t6n tr6m mang theo chi6c t[i c6 kich cd ld p e Nx. V6y h6n ph6i chon lda mdt danh siich ciic db vdt s6 mang di nhd th6 nio dd cho tdng gi5 tri t6y c5p dddc ti t6n nhdt. Tfc: Tim x[1], x[2],,,,, x[n] (vdi x[i] e N : sd lLtdng loai db v6t thf icen ldy) sao cho: ,; Ix[i].mlil < p ve lx[i].c[i] dat gi5 tri ctrc dai x Input: + n ld sd d6 v6t, kich thd6c cdc db khSc nhau. Loai d6 v6t i 1 2 n Kich cd mlil mlll ml2l mlnl Gi5 tri clil cl 1l cl2l clnl Vdi m[i] e N' , c[i] e R + p lA cd trii x Output: Tim (X[1], X[2] . . . X[n]) (X[i ] :sdtoaidbv6t i, i=1 n) fZ,lfulil= p 5ao cno: tI'*t,i.t,t-+ max n Loai db v6t i 1 2 3 4 5 Kich cd mli 5 6 3 7 B Gra tn cl I 2 5 2 6 1 r' Vidt1: n = 5 ; p = 10, dtJdccho x K€t guir: + Sd db virt 5, kich thd6c 1"0 + Tdng gi5 tri nh6p dr/dc vdo t(i xSch: B + Sd lrrOng ldy theo th[r tlr nhd sau: 0 0 1 1 0 b. Ph6n tich bii to6n: Brl6c 1: Ph6n tich bii toSn: Goi P(r, s) li bdi todn chi6c tfi x6ch, vdi: r e Nx: kich cd chi6c t[i s e N*: s6 c5c loai d6 v6t khdc nhau (bdi todn ban d6u td p(p, n)) C5c gi5 tri cEn tim: t[r,s]: gi5 tricdc dai Ixlil,c[i] cia bditoSn p(r, s) [r,s]:_s6 h/dng loaidb v6t s t6i uu c6n l6'y (tr?c: x[s]) ctia bditodn p(r, s) Brfdc 2: ciii phSp ct€ quy: Khis=1: u[r, 1] = r div m[1] Ilr, 1l = u[r, 1]*c[1] Khis>1: l[r, s] = max { k*c[s] + lfr - kxmfs], s-11 ] 0<k<rdivm[s] = koxclsl + lfr - h*mls], s-11 ufr, s] = ks Brldc 3: LQp bing Procedure LapBang; Begin For s:=1 to n do For r:=0 to p do If s=1 then Tinh u[r, 1] vir l[r, 1] else Tfnh u[r, s] vd l[r, s]; End; Vi du s R 1 2 3 4 5 o o\o o\0 o\o 0\0 o\0 1 0\o 0\o o\o o\0 o\o 2 0\0 0\0 0\0 o\0 0\0 3 0\o 0\o 2\1 2\o 2\O 4 o\o 0\o 2U 2\o 2\0 5 2\1 2\O 2\o 2\o 2\o 6 2U s\1 s\o s\o s\o 7 2\1 su s\o 6\1 6\o 8 2U s\1 s\0 6U 6\0 9 2\1 s\1 7\1 7\0 7\0 10 4\2 su 7\1 8U 8\0 x x€'t qui: + Sd d6 vat s, kicfr thrrdc to + Tdng gi6 tr! nh6p dUdc vdo tfixiich: B + Sd luOng ldy theo thf/ tLr nhut sau: 0 0 1 1 0 D0 phrirc tap tinh to6n: O(np2) BrJ6c 4: Tdng hqrp k€t qui Procedure TongHop; Begin r: =pi For s:=n downto 1 do Begin x[s]:= u[r,s]; r:= r - x[s],mfsl; End; End; E6 phr?c t?e tinh to6n: O(n) BrJ6c 5: Cei dit chddng trinh Program TuiXach; Uses crt; Var n,p:word; x,m:arrayf1., 10] of word; c:arrayf1 10] of real; l:array[0 10,1 10] of real; u:array[0 10,1 10] of word; { } Procedure NhapDL; Var i:byte;f:text; Begin assign(f,'tuixach.inp'); reset(f); read(f,n); readln(f,p); End; { } Procedure Lapbang; Var r,s,k,k1 :word; max,tam : real; Begin for s:=1 to n do for r:=0 to p do if s=1 then Begin u[r,1]:= r div m[1]; l[r,1]:=uIr,1]*cl1]; End else Begin k1: =0; max:= 0; for k:=0 to r div m[s] do Begin for i:=1 to n do read(f,mlil); for i:=1 to n do read(f,c[i]); writeln('So vat ',n,', kich thuoc lieu trong file tuixach.inp'); close(f); tui 'rp,', du Tam : = kxclsl +lIr-kxm[s],s-11; if tam>max then Begin Max:=tam; k1:=k; end; end; lIr,s]:=max; u[r,s]:=k1; End; End; { } Procedure tonghop; Var r,s:word;f:text; Begin assig n(f,'tuixach.out'); rewrite(f); r:=pj for s:= n downto 1 do Begin xlsl: =uIr,s]; r:=r-x[s]xm[s]; End; End; { } Procedure Inkq; Var s:word; f:text; Begin assig n(f,'tu ixach.out' ); rewrite(f); writein('Tong gia tri nhan duoc rrao tui xach la: ',1[p,n]:4:1); write('So luong lay theo thu tu nhu sau: '); for s:=1 to n do Begin write(f,xIs]:3); write(x[s]:3); End; writeln; close(f); End; { } Procedure Inbang; Var srr:word; Begin writeln; for s:=1 to n do write(s:S,' '); writeln; for r:=0 to p do Begin write(r); for s:=1 to n do Begin write(lIr,s] : 3 : 0,'\', u Ir,s],"); End; writeln; end; End; t ) BEGIN Clrscr; NhapDL; Lapbang; Tonghop; inbang; Inkq; Readln; END. { } BiitQp 2: ph6p nhAn td hdp nhi€u ma trQn (O", ) \J a. PhSt bidu bii to6n CEn tinh M = M1xM2x ,xMn Trong d6: M, Id ma tr6n c5'p m[i-1]xm[i] (i=1 n) . H5y xdc dinh thf tUthr/c hi6n cdc ph6p nh6n sao cho s6 ph6p tinh Ie t6ithidu. x Input: + C6'p cr]al ma tr6n Vdi m[i] e N x Output: X5c dinh thf td tinh: sd ph6p tinh > min xVidg:TinhM= M1 * MZ " M3 | 10x201[20xs0] [s0xs] (MrxM2)xM, c6 sd ph6p todn td: 10x20x50 + 10x50x5 = 12500 M1x(M2xM3 ) c6 sd ph6p todn ld: 10x20x5+20x50x5=6000 x K€'t qui: M1 x M2 x M, c6 sd ph6p todn lir: 6000 b. PhAn tich bii to6n: Bddc 1: Ph6n tich bii toin: Goi P(r, s) ld bdi todn nhdn ma tr6n: MrxMr*rx xM' vdi r < s cua n ma tr 0 1 n mlil ml0l mlll mlnl (bai todn ban d6u td p(1, n)) Gi5 tric6n tim: !lt,r]t vitri ph6p to5n thrrc hi6n cudi cing cia beitodn p(r, s) (MrxMr*1x xMp-1) x (MpxMp* 1" rM5) k = k[r,s] e [r+1, s] [r, s]: sd ph6p tinh nh6n t6i uu cira baitodn p(r, s) BtI6c 2; ciii ph5p dG quy: Trr/o'ng hdp 1: Khis = r: l[r, r] = g Trr/dng hgp.2t KhL5 = p1g. lt r,Q t\< hf'- r fx m Jf.1 J I q 1 kfqv+11 =-1+1 Trddng hdp 3r Khi s > r<S: l[r, s] = min { l[r, v-1] + m[r-1]*m[v-1]*mlsl + lfv, sl ] r+1 <v<s (v lA c5c vi tri ph6p todn thdc hi6n cudi cirng khdc nhau) = l[r, vL1] + mlr-11 * mlvll] {, m[s] + l[v', s] (v' ld vi trit6i rru trong sd cdc vi tri cira v) k[r, s] = Y' Nh6n X_6t: C6 thd gh6p trddng hdp 2 vAo trddng hdp 3l Brfdc 3: Lflp bing Procedure LapBang; Begin For s:=1 to n do For r:=s downto 1 do If r = s then lfr, r] := 0 End; else Tinh l[r' s] vi k[r' s]; Vi^!U: Tr/ vi du tr6n ta l6p dddc bing sau 56 ph6p to6n tdi thidu td: 6000 Thf t{thrJc hi6n ph6p nh6n: 3 2 r s 1 2 3 1 0lt 2 10000/2 012 3 600012 s000/3 013 D6 phr?c t?p tinh to6n: O(n3) Brf6c 4: Tdng hep k6t quJ Procedure TongHop; Begin writetn('Sd ph6p tinh le t6i thidu:,, l[1, n]); i:=r'ti 1-imW(1, n); {NhEm lEn [fdt xiic dinh cdc gi5 tri x[i]: vi tri ph6p nhSn cEn thuc hi6n trong tEn nh6n thf i(i=1 n-1)) writeln('Thr? t{ thttc hi6n cdc ph6p nh6n :,); For i:=1 to n-1 do writeln(xfil); End; Trong d6 th[r tuc TimW dr/dc x6y dqfng d9 quy nhf sau: Procedure TimW(r, s); Begin If (r < s) then Begin i:=i-1; vt:= k[r, s]; xlil:= vt; TimW(r, vt-1); TimW(vt, s); End; End; Bddc5: CAi d{t chtldng trinh Program Matran; Uses Crt; Var n:word; m :array[0 20] of word; x:arrayf1 20] of word; k,l :arrayf1 20,1 20] of integer; { } Procedure NhapDl; Var i:byte;f:text; Begin assign(f,'matran, inp'); reset(f); readln(f,n); for i:=0 to n do Begin read(f,mIi]); write(mIi]:3); End; writeln; close(f); End; { } Procedure Lapbang; Var r,srvrvmax:word; Begin for s:=1 to n do for r:=s downto l" do if s=r then Begin lIr,r]:=0; klr,rl:=r; End else Begin VmaX:=f + 1; l[r,s]:=l[r,r]+mIr- llxmIr]xm[s] +lIr+ 1,s]; for v:=r+2 to s do if l[r,s]>l[r,v-1]+m[r-1]xm[v- llxm[s]+l[v,s] then Begin VMaX:=V; llr,sl : = lIr,v- 1] +mIr-1] xmlv- llxm[s]+l[v,s]; End; k[r,s]:=vmax; End; End; { } Proced ure timvt( r,s :word) ; Var vt,i:word; Begin if r<s then Begin i:=i-1; vt:=k[r,s]; xlil:=vt; timvt(r,vt-1); timvt(vt,s); End; End; t ) Procedure tonghop; Var i:word;f:text; Begin assign(f,'matra n.out'); rewrite(f); writeln('so phep toan toi thieu la: ',1[1,n]); writeln(f,'so phep toan toi thieu la: ',1[1,n]); i:=rtj timvt(1,n); write('thu tu thuc hien cac phep nhan:'); writeln(f,'thu tu thuc hien cac phep nhan:'); for i:=1 to n-1 do Begin write(xIi]:3); write(f,xIi],' '); End; Close(f); End; { } Procedure Inbang; Var s,r:word; f:text; t Begin writeln; l for s:=1 to n do write(s:10,' '); -j writeln; for s:=1 to n do Begin write(sr' '); for r:=1 to s do t Begin write(lIr,s]:5 ,'f' ,klt,s],' '); End; writeln; end; End; { } BEGIN Clrscr; NhapDL; Lapbang; Tonghop; Inbang; Readln; END. Bii tdp 3: Tim x6u trong cdc dai. a. PhSt bidu bii to6nr Bei toSn x6u t1o1s cr"lc dqi: s ld x6u trong cia T n6u s nh6n dddc bSng cSch xo5 di m6t sd t<f ttl nAo d6. Vidu: 'ABC, ld x6u trong cria.GAHEBOOC, Bei todn: cho 2 x6u T1, T2. Tim m6t x6u S ld x6u trong chung cria T1 vd T2 co d6 ddi ct;c dAi, '-"r -"errJ vev 'r ve Vi du: TI='ABCDAE'vd T2='XYACADK' c6 x6u 'AcD' le x6u trong chung vdi d6 ddi cdc dai. * Input: X6u T1 vd T2 * output: X6u s li x6u trong chung cira T1 vir T2 c6 d6 dei cdc dai b. PhAn tich bii tof n: Budc 1: phin tich bii toin: Goi n lA d6 dei cria x6u T1 m ld d6 ddi c[ra x6u T2 Ta cEn xdc dinh x6u con chung S ddi nh6't c0a 2 x6u T1 vd T2 Gi6 sir L(i,j) c6 d6 dei x5u con chung ldn nh6't crla z x6u T1 c6 d6 dii i vd x6u T2 c6 d6 ddij (i<=n, j<=m). Btt6c2= ciii ph6p d6 quy: + Ndu i=0 thi L(0,1;:9, Ndu j=g thi L(i,0)=0 + Ndu i>0 vd j>O va T1;=72-thi L(ij)'= i + L(i_1j_1) + N6u i>0 vd j>0 vd T1i<>T2j tni L(i;l = rvaxiLli,j_rj,l1i_r,i;; Brf6c 3: L6p bing; Procedure LapBang; Begin For i:=1 to n do L[1,0]; =0; For j:=1 to m do L[O,j]: =0; For i:=1 to n do For j:=1 to m do if T1[i] = T2[] then L[i,j]:= 1 + Lli_1,j_11 else L[i,j] := maxab(L[i,j_1], L[i_1,j]); End; Vi du: cho hai x6u T1='ABCDAE'vd T2-'XyACADK,. Tim x6u con chung c6 d6 dii rdn nh6.t. BA I L. * c.D 4 , G , \T; T1\ J\ A TZ I A a C (n A a D C K l 1 A 0 0 t€ I I 1 t B 0 0 lF I I I I (- 0 0 I oz w'l 2 2 D 0 0 I 1 L 2 3@ T*3 A 0 0 1 2 3 J ?= ,& Kdt qu6 x6u chung ddi nh6't ld: ACD D0 phr?c tap tinh to5n: O(n.m) Brl6c 4: Tdng h{p k6t qui: Procedure Tonghop; Begin i:=hi j:=m; S:="; While (L[i,j]>0) do Begin if T1[i]=T2Dlthen Begin S:=T2Ul + S; :._; {, l,=1;' End else if L[i, j]= Lli-1, jl then i:=i-1 else j: =j- 1; End; Writeln('Xau chung dai nhat:', S); End; D6 phfc tap tinh to6n: O(n). Bd6c 5: Cii det chddng trinh Program XauChung; uses crt; Var L:Array[0 100,0 100] of Byte; s1,s2,kq: String; m,n: Byte; { } Function maxab(a,b: byte) : byte; Begin if a>b then maxab: =a else maxab: =b; End; { } Procedure NhapDL; Var f : Text; Begin Assign(f,'xauchung, inp'); Reset(f); Readln(f, s1); Readln(f, s2); m:=Length(s1); n: =Length(s2); Close(f); End; { } Procedure LapBang; Var i,j: Byte; Begin For i:=1 to n do lIi,0]:=0; For j:=1 to m do ll0,jl:=0; For i:=1 to m do For j:=1 to n do if sllil = s2ljl then L[i,j]:= 1 + L[i-1,j-1] else L[i,j]:= maxab(L[i,j-1] 1,il); End; { } Procedure Tonghop; Var i,j:byte; f:text; Begin Assi gn(f,'xauchung. out' ); Rewrite(f); i: =mi j: =n; kq:="' While (L[i,j]>0) do Begin if s1[i]=s2Ulthen Begin kq:=s2[] + kq; i:=i-1; j:=j-L; End else if L[i, j]= L[i, j-1] then , L[i- j:=j-1 else i:=i-1; End; Writeln(f,'s1 - ',s1), Writeln(f,'s2 = ',s2); Writeln(f,'Xau chung dai nhat;', kq); Close(f); End; { } Procedure Inkq; Begin Writeln('s1 = ',sl)i Writeln('s2 - ',s2), Writeln('Xau chung dai nhat:', kq); End; t ) i Procedure Inbang; var i,j:byte; Begin \ writeln(' LAp BANG \; write("); for j:=1 to m do write(s2[]:4); writeln; for i:=1 to n do begin wrlte(s1[i]); for j:=1 to m do write(lli,jl:4); writeln; end; End; { } BEGIN Clrscr; NhapDL; Lapbang; Tonghop; Inbang; Inkq; Readln; END. \ a. phSt bidu bii to6n: Bii tQp 5; Bii to5n ngr/di du lich M6t ngfdi di tt/ thdnh phd 0 ddn thdnh ph6 n vd c6 thd di qua n-l thanh phd kh5c 1, 2,.,., n- 1, theo 16trinh: 0-+ i1 - i2 _+ i;,-+ n, trong d6: O < t i t. ik. n, Gi5 v6 c0a xe ditr/thdnh phd id6n thdnh ph6 j ta cgi,i1. Tim m6t 16 trinh ttJ thdnh phd 0 d6n thdnh pr'ro n-rul"."r.)o tdng chi phf v€ gi5 v6 dat cdc tidu,* Input: +:- ld thtinh ph6xudt phdt lrrr vrrr vs vro + s li dfch + Ma tr6n cli,jl (i,j =0 n) tuu trii chi phf x output: L6 trinh ngSn nh6't trr thdnh prr6 o o8n tnirnh phd n. b. Ph6n tich bii todn: Brl6c 1: Ph6n tich bii to6n: x Goi P(r,s) lA bei todn du lich, vdi: - r e N: dinh xudt phdt - s e N: dinh cEn d6n (bei todn ban dEu: p(0,n)) x C5c gi5 tri c6n tim: - l[l,s] : chi phi nhd nh6't d6 di ti/ r _> s c0a bei todn p(r,s). - ufr,sl : dinh k6 cu6i tr6n dL/dng di tf r-> s cira odi toJn p(r,s). BtI6c2= Giii ph6p d€ qui: -Khir=s: l[r's] = 6 u[r,s] = -1 -Khi r + s: l[r,s] = min(l[r,k] + c[k,s]) = l[r,ko]+c[ko,sJ (r-<k<s) ulr,s] = ko Brfdc 3: LAp bing: Procedure Lapbang; Begin Fors:=0tondo Forr: =0tosdo Ifr=sThen Begin l(r, r): = 0; u(r, r) = -1' End Else Tinh l(r,s)) vir u(r,s); End; Dfdng di tU thenh phd 0 ddn thdnh ph6 4 c6 t6 trinh: 0 >1 >2 >4 Chi phi nh6 nhdt dd di tiJ thdnh phd 0 ddn thdnh ph6 4: 15 D6 phr?c t?p tinh to6n: O(n3) Brl6c 4: Tdng hdp: Procedure Tong hop(d,c :word ); Begin i:=0; x[0]:=c; k:=ci while k<>d do Begin i:=i+1i xlil:=u[d,k]; k:=u[d,k]; End; End; D6 phr?c tap tinh to5n: O(n) Brl6c 5: Cii d{c chr.tdng trinh Program DuLich; Uses cft; Const max=100; Var n:word; x:array[0 max] of integer; c,l,u:array[O max,O max] of integer; { } Procedure NhapDL; Var i,j:byte;f:text; Begin assign(f,'C: \d u lich. inp'); reset(f); readln(f,n); for i:=0 to n do for j:=0 to n do read(f,cIi,j]); close(f); End; { i Procedure Lapbang; Var r,s,k,k0:word; Begin for s:=0 to n do for r:=0 to s do if s=r then Begin lIr,s]:=0; u Ir,s]: =- 1; End else Begin k0: =s-1; lIr,s] : = lIr,s-1] +c[s- 1,s]; for k:=r to s-1 do if l[r,s] > l[r,k]+c[k,s] then Begin k0: = k; lIr,s] : = lIr,k] +c[k,s]; End; u[r,s]:=k0; End; End; { } Procedure tonghop(d,c:word); Var i,k,j,t:word;f:text; Begin assig n(f,'C: \dulich.out'); rewrite(f); i:=0; s r o 1 2 3 4 0 o -1 1 7 0 o -1 2 L2 1 5 1 0 -1 3 L2 0 7 1 4 2 0 -1 4 15 2 8 2 3 2 4 3 o -1 x[0]:=c; k:=c; while k<>d do Begin i:=i+1; x[i]:=u[d,k]; k: =u[d,k]; End; . writeln(f,'Duong di tu Tp ', d,, den Tp ',c,, chi nho nhat la:'); for j:=i downto 0 do . Begin write(f,x[]); if j>0 then write(f,' >,); End; writeln(f); write(f,'chi phi la: ',1[d,c]); close(f); End; { } Procedure Inkq; Var f:text; ch:string; Begin Assign(f,'C: \DuLich.Out'); reset(f); readln(f,ch);write(ch); readln(f,ch);writeln(ch); read I n(f,ch);write(ch); close(f); End; { } Procedure InBang; Var i,j:Byte; Begin phi for i:=0 to n do write(i:g);writeln; for j:=0 to n do Begin write(;':4,' '); for i;=0 to j do write(uIi,j],'/,,1[i,j],' '); writeln; End; End; t ) BEGIN Clrscr; NhapDL; Lapbang; Tonghop(0,4); Inbang; Inkq; Readln;END. a. phSt bidu bai toSn: Bia tip 5: Bii toSn sinh vi6n 6n thi, Bei todn sinh vi6n 6n thi: M6t sinh.vi6n cdn T ngdy dd 6n thi n m6n. Theo kinh nghi6m cia anh ta, ndu 6n m6n j trong i ngdy thi drdc oidr'H J1i,;1. cia stJ cho nia .a. a[i,j] (vdi a[i,j] < =a[i+ 1,j]). Tim b6 xUl (sd ngdy 6n m6n j, v6i j=1,.n) sao cho Ix[1=, vd sinh vi€n dat tdng didm idn nhdt (IalxUl, jl-+max). x Input: + n ld sd m6n + m td s6 ngdy + aIij] didm crla 6n thi m6n j trong i ngiry (ali,jl<=a[i+1,j]) x out put: -,, r"J,]rJiT;';f; i'r,=I n) *o *orx[]=, va sinh vi6n dat tdns didm rdn nhdt (IalxUl, jl-+max), b. Ph6n tich biito5n: Brldc 1: ph6n tich bii to6n: x Goi p(r,s) ld bii toiin SV 6n thi r m6n trong s ngEy (bai to6n ban d6u s6 ld : p(n,m)) " * Cdc gi5 tri cEn tim lir: bait"drl[i;rrj.tunn sd didm ldn nh6't mir sinh vi6n c6 thd dat dudc khi 6n thi r m6n trons s ngdy cria _ ufr,gJ : 56 ngdy tdi uu cEn d6 6n m6n r c0a bii todn p(r,s). Bttdc2: ciii phip dQ qui: -flrif = 1:l[1,r] =a[s,1], u[1,s] = I -Khi i> 1:l[r,s] = max(attit it;r-r,s_t1; ) 'fi<t<-sl a[h,r] + llr-1, s-ksl Brf6c 3: L6p bing: u[r's] = ko Procedure Lapbang; Begin for s:=1 to m do for r:=0 to n do End; if (s =.1) then Tinh lf1,sl vd ufl,sl else Tinh l[r,s] vi u[r,s] vf du: c6 3 m6n thi 6n trong 5 ngdy vir didm cdc m6n tr/dng r?ng cdc ngdy nhf sau: . v6t i 1 2 3 4 5 Kich cd mli 5 6 3 7 B Gra tn cl I 2 5 2 6 1 r' Vidt1: n = 5 ; p = 10, dtJdccho x K€t guir: + Sd db virt 5, kich thd6c 1"0 + Tdng gi5 tri nh6p dr/dc. vdi: r e Nx: kich cd chi6c t[i s e N*: s6 c5c loai d6 v6t khdc nhau (bdi todn ban d6u td p(p, n)) C5c gi5 tri cEn tim: t[r,s]: gi5 tricdc dai Ixlil,c[i] cia bditoSn p(r,. Output: X5c dinh thf td tinh: sd ph6p tinh > min xVidg:TinhM= M1 * MZ " M3 | 10x201[20xs0] [s0xs] (MrxM2)xM, c6 sd ph6p todn td: 10x20x50 + 10x50x5 = 1 250 0 M1x(M2xM3