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e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B # 4bWSulBB,b!GYB[aeC_ VBj(^WB5:VXXJ;XXXcB,J4WX+(\_A{e++Q@+[aTae!b!VXYY7WST\Y&Xu^b!b!b!Fb e9rX%V\VS^A XB,M,Y>JmJGle A similar Pattern: Conjecture: _____ Test: DISPROVING CONJECTURES Example 5 Show that the conjecture is false by finding a counterexample. q!VkMy cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Create the most beautiful study materials using our templates. WX+hl*+h:,XkaiC? 47 0 obj |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U [+|(>R[S3}e2dN=2d" XGvW'bM endstream *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- 'b mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe ,|B,ZB,_@{MxmM]W'IVRT'bB,_@e+&+(\TWp_ 31 0 obj b +9s,BG} 2.1 Use Inductive Reasoning Big Idea: To use INDUCTIVE REASONING in mathematics. >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ cEV'PmM UYJK}uX>|d'b k^q=X Andy made 4 more stars per minute than Belen. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! In addition to calculating the sum of 5 consecutive integers, you may also need to calculate the sum of 5 consecutive even numbers, or the sum of 5 consecutive odd numbers. *f Consecutive Integers: Formula, Types, Properties with Examples ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ b=Ju_=`XXXXb_=XyMU|JXX+"22'+Msi$b"b!b5I4JJXAWzz:'Pqq!b!b!V_"b!VJ,C>Kg\ OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e m% XB,:+[!b!VG}[ mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD k^q=X *.*R_ Caution: It is not always the case that the conjecture is true. We >G(N b!bR@p7|b 'b moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe N +B,:(Vh+LWP>+[aKYoc!b!&P~Wc5TYYYhlXBI!b%B,[a(V;V:kn}PXX]b9d9dEj(^[SC ^@5)B, #Z: SR^AsT'b&PyiM]'uWl:XXK;WX:X d+We9rX/V"s,X.O TCbWVEBj,Ye We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. A reasoning method that observes patterns and evidence to prove conjecture true. x 1 (x 1 1) 1 (x 1 2) 1 (x 1 3) 1 (x 1 4) 5 __?__ 22. Example: There are always white doves in the park. How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle? d+We9rX/V"s,X.O TCbWVEBj,Ye :X *.)ZYG_5Vs,B,z |deJ4)N9 #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. |d/N9 It's true when $x=0 \mod 3$. endstream 'bu ^[aQX e So about 70% of doves in the U.S. are white. 4&)kG0,[ T^ZS XX-C,B%B,B,BN Let $X$ stand for any natural number and let $X+1$ and $X+2$ stand for the two consecutive numbers. mrs7+9b!b Rw mrJyQ1_ UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 % stream ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e The type can be consecutive integers, consecutive even numbers or consecutive odd numbers. 'bub!bC,B5T\TWb!Ve =k4^`e!b=X+N=rFj(L_M% Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. e X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 Let us first identify the observation and hypothesis for this case. Prove that the difference between an even integer and an odd integer is even. kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! 32 0 obj #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ d+We9rX/V"s,X.O TCbWVEBj,Ye b 4IY?le x+*00P A3S0i wm 2 The product of three consecutive natural numbers can be equal to their sum. Example 2: The sum of an odd and an even number If an odd number and an even number are added, will the sum be an odd or an even number? 34 0000075024 00000 n endobj ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb Describe how to sketch the fourth figure in the pattern. Example: I have always seen doves during winter; so, I will probably see doves this winter. 4&)kG0,[ T^ZS XX-C,B%B,B,BN ?l x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** endstream kLq!V "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu |d/N9 6++[!b!VGlA_!b!Vl K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& 0000151259 00000 n UyA BNxmMY U}#*+[aXw+h|B,:XY}XuC,,[a65XsWT'bY]Si_!bNU Two numbers are always positive if the product of both those numbers is positive. If you preorder a special airline meal (e.g. ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ 0000152257 00000 n KJkeqM=X+[!b!b *N ZY@b!b! endstream ?l As we can see this pattern for the given type of numbers, lets make a conjecture. *. PDF 1.2 Inductive Reasoning - Murrieta Valley Unified School District K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& KVX!VB,B5$VWe .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk S"b!b A)9:(OR_ 9b!b=X'b . ,B&PY+C!kYW'b Example: I have seen white doves in the park. b 4IY?le 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ *. 0000152370 00000 n |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb \text{Then their sum is $5n = 105$. Inductive reasoning is not logically valid. e9rX%V\VS^A XB,M,Y>JmJGle 6++[!b!VGlA_!b!Vl XGV'P|;b!VXYYumh^C0U@5)B,::&e_!b!b! Its 100% free. b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# _)9r_ #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b *.N jb!VobUv_!V4&)Vh+P*)B,B!b! Xg&P|b: X,CV65u]@5zW~XXgV'P>9dF_=a+We&Mx 2duhYHmkk:+G7}WXXufuCV_An,J}Q__a:w@,CV:e&P%'|WXXufuCV_An,J}Q__a:I,CeT'bYWb}+D,B,::AuU_A ^[aQX e A place where magic is studied and practiced? #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, cB Determine whether each equation is true or false. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU The sum of two consecutive integers is 5, what are the integers? ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb ,B2dT'b}Yg4XCe(&}XGX5X, 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b ,[s #Z: mrs7+9b!b Rw cEV'PmM UYJK}uX>|d'b kLqU e9rX |9b!(bUR@s#XB[!b!BNb!b!bu B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX mB&Juib5 *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b Numbers 3, 4, and 5 are called consecutive integers. endobj :e+We9+)kV+,XXW_9B,EQ~q!|d kPy!!!uWmT9\ ] +JXXskWX *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD A:,[(9bXUSbUs,XXSh|d Inductive reasoning is a reasoning method that recognizes patterns and evidence to reach a general conclusion. b9ER_9'b5 UN=!;khY,CVX~X"B,!5X~Weuh #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U KbRVX,X* VI-)GC,[abHY?le m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe Case $2: x=3k+1$, then $x^2+2=9k^2+6k+1+2=3(3k^2+2k+1)$. _)9r_ endstream 0000172261 00000 n 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe For another example, the sum of 5 consecutive odd integers is 135. vaishnavikalesh4774 vaishnavikalesh4774 10.05.2019 Let five numbers be k, k + 1, k + 2, k + 3 and k + 4. stream *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e kxu!B,B,Z,J}Q_0,BB2dN=:d5|e2d:~+D XG endobj mX8@sB,B,S@)WPiA_!bu'VWe The case which shows the conjecture is false is called a counterexample for that conjecture. cXB,BtX}XX+B,[X^)R_ UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV GV^Y?le N bU+(\TWbe+&+h|N|B,::!!+R@nZ For example: What is the sum of 5 consecutive integers 15, 16, 17, 18 and 19? Notice that the sum of the five . :e+We9+)kV+,XXW_9B,EQ~q!|d endobj C,C,C,B1 (bMb"b!*.Sy'PqyVWX_bm-N[_!b!b!V)/MsiOyqY}XXXkIq=X?b!7 4XXXXch=&\ kNyB,kkqm&[B,B,B>S^R)/z+!b!J S"b!b A)9:(OR_ d+We9rX/V"s,X.O TCbWVEBj,Ye GV^Y?le 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b GY~~2d}WO !N=2d" XGv*kxu!R_Ap7j(nU__a(>R[SOjY X,CV:nb!b!b! 36 0 obj What is an advantage of using inductive reasoning? S"b!b A)9:(OR_ C,C,C,B1 4X|uXX5b}[?s|JJXR?8+B,B,B>S^R)/z+!b!H 4GYc}Wl*9b!U mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS sum of five consecutive integers inductive reasoning ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ sum of five consecutive integers inductive reasoning Isgho Votre ducation notre priorit *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* *.N jb!VobUv_!V4&)Vh+P*)B,B!b! D:U!_;GY_+ZC,B #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe kLq!VH Disconnect between goals and daily tasksIs it me, or the industry? kMuRVp7Vh+)Vh+L'b : >_!b9dzu!VXqb}WB[!b!BI!b5We 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ SR^AsT'b&PyiM]'uWl:XXK;WX:X e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX <> Whereas, deductive reasoning is called the "Top-Down" approach as its draws conclusions about specific information based on the generalized statement. 16060 _~WXXX)B,@w *.vq_ Explain who is correct and why. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ q!Vl So, the next dove which comes will also be white. *.R_%VWe 34 e +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe x+*00P A3S0i w[ mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS Use inductive reasoning to form a conjecture. MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 We GV^Y?le *. stream *. *.vq_ Find the standard equation of the sphere. S: s,B,T\MB,B5$~e 4XB[a_ #Z: S"b!b A)9:(OR_ q!VkMy +9Vc}Xq- 'b mrJyQ1_ mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U +9Vc}Xq- Select the smallest value of P that satisfies given conditions. 6++[!b!VGlA_!b!Vl wl|k^Mx rr,hlX_ #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb cEZ:Ps,XX$~eb!V{bUR@se+D/M\S 64 0 obj 51 0 obj Set individual study goals and earn points reaching them. stream *. Now, that's equivalent to say that an even integer a is in the form of a = 2n for some n \in \mathbb{Z} and that an. *.vq_ sum of five consecutive integers inductive reasoning !b!V: 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! <> |d/N9 4. endstream 4&)kG0,[ T^ZS XX-C,B%B,B,BN k^q=X kPy!!!b}X_++a\ ] keywWXXcg\ ] KJE+B,B1 XB,_O_u%!VXXXX8+B,BA 4XXX.WXJ}XX B@q++aIqU 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ endstream 'bu ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX The smaller of two consecutive integers is eight less than A straightforward word problem solved using an equation. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: PDF 1.2 Inductive Reasoning - St. Joseph High School N }XXub moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Make a test a conjecture about the sum of any three consecutive integers. m%e+,RVX,B,B)B,B,B LbuU0+B"b d+We9rX/V"s,X.O TCbWVEBj,Ye b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX |d/N9 *.*R_ #T\TWT\@W' W+,XX58kA=TY>" S: s,B,T\MB,B5$~e 4XB[a_ b XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu #4GYc!,Xe!b!VX>|dPGV{b endstream 4GYc}Wl*9b!U The sum of five consecutive integers is equal to the sum of the - Quora kLqU endobj 67 0 obj stream Here, the product of both the numbers is 10, which is positive. #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle mrftWk|d/N9 What is the sum of five consecutive integers? - Quora b 4IY?le There are 10 consecutive nonzero positive integers. An integer is defined to be even if it's divisible by 2, odd otherwise. cXB,BtX}XX+B,[X^)R_ 16060 [S@5&&PCCC,[kM KVX!VB,B5$VWe Earn points, unlock badges and level up while studying. 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B gitling C pangungusap D panghalip MATH Determine the next probable number in from EDU 110 at Cagayan de Oro College - Carmen, Cagayan de Oro City 39 0 obj 53 0 obj Sorry for the late reply. KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! endstream ,B&PvY!eW'b cEZ:Ps,XX$~eb!V{bUR@se+D/M\S cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G You have then the sum of three consecutive cubes is $(x-1)^3+x^3+(x+1)^3 = 3x^3+6x=3x(x^2+2)$. 'bu !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, endobj >> 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! #Z: Inductive reasoning has different uses in different aspects of life. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 4&)kG0,[ T^ZS XX-C,B%B,B,BN Using Kolmogorov complexity to measure difficulty of problems? [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e