1. 1 http://mathhmagic.blogspot.gr/ 124

2. 2 http://mathhmagic.blogspot.gr/ 124 , ... , . , . , - . . !!. ( ) 1) Henry Ernest Dudeney - 536 Puzzles and Curious Problems 2) Phillips, Hubert 1961. My best puzzles in logic and reasoning. Dover, NY. 3) Phillips, Hubert 1962. My best puzzles in mathematics. Dover, NY 4) Joseph S. Madachy Madachy's Mathematical Recreations 5) Martin Gardner, My Best Mathematical and Logic Puzzles ( Martin Gardner ) 6) Aaron J. Friedland, Puzzles in Math and Logic ,Dover 7) Stephen Barr ,Mathematical Brain Benders ,Dover 8) Sam Loyd ,Mathematical Puzzles of Sam Loyd ,Dover 9) Boris A. Kordemsky,The Moscow Puzzles ,Dover 10) D. O. Shklarsky , The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics 11) R. Todev, International Mathematics Tournament of the Towns 12) Paul Vaderlind ,The Inquisitive Problem Solver (MAA Problem Book Series) 13) Martin Erickson Aha! Solutions (MAA Problem Book Series) 14) G. Polya ,The Stanford Mathematics Problem Book ,Dover 15) Paul J. Nahin , Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction 16) David Wells, Games and Mathematics: Subtle Connections 17) Raymond Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles 18) Raymond Smullyan, Satan, Cantor and Infinity : http://mathhmagic.blogspot.gr/ 3. 3 http://mathhmagic.blogspot.gr/ 124 ( Raymond Smullyan ) 19)A.Engel , Problem solving strategies 20)D.Fomin, S.Genkin,I.Itenberg,Mathematical circles( Russian experience) 21)P.Zeitz, The art and craft of problem solving 22)Barbeau Ed, Klamkin M ,Five hundred mathematical challenges 23) , 24) Kvant ,Crux Mathematicorum ,American Mathematical Monthly, , , , 25) ,.. 26) , .- . 27) 1,2,3, Bolt,B 28) , . 29) , . 30) , 31) David Darling, The Universal Book of Mathematics 32) Barbeau .,Mathematical Fallacies,Flaws, and_Flimflam_(MAA_2000) 4. 4 http://mathhmagic.blogspot.gr/ 124 Sam Loyd Sam Loyd Uppingkton . " ". Uppingkton 110 . 3000 Uppingkton . 15 . Loyd . Loyd : 1. 6 . 5. 5 http://mathhmagic.blogspot.gr/ 124 2. !!! . . . . , , , . . , , . , . : ( , ) . , , , . ( ) . ; 3. . ( . 21,22,23) , 60. 3 100 ( . 18). ( ) 67 (84x 67=5628). ( ) . ; 3 3 1: 18:3+1=7 28-7=21 . : 2x28=56. ; 6. 6 http://mathhmagic.blogspot.gr/ 124 4. !!! . . ( ) . . Tournament of towns 1989 5. .. !! , . , ( .) : (1) . (2) . : (3) . (4) . : (5) . (6) . : () ( ) () . ; 7. 7 http://mathhmagic.blogspot.gr/ 124 6.!! , 5 (), (), (), () () 5 , V,W,X,Y . 5 1 ,2 ,3 ,4 , 5 5,4,3,2, 1 . . ( 5 ) 24 , 4 5 , 4 V, 5 W 3 . . 5 . 7. !! : = 30+40 cm =40+30 cm =50+ cm , ,, . (,,) . 8. . !!!! - . . , . 1 . . , ; 9. : !! . : PASSION KISS= 0-9. , . . 8. 8 http://mathhmagic.blogspot.gr/ 124 10. , - , . . , : , , . . (Kvant magazine) , Sam Loyd! O Leonard Euler, . ; Euler , " ". Sam Loyd, 14-15 , Euler . " " . Scienc American . Loyd , fortran 1976 . Loyd : " . , , , , , , , , , . , , 8 . , , ." , : , , , , , , , , ( ) 9. 9 http://mathhmagic.blogspot.gr/ 124 ;; , , . . 10% 10% . ; , !!! 1000 . , 10% 100 1100 . 10% 10% 1100 , 110 990 . , 10% 10% . 100 , ( ) . 100 45 55 . 10000 . , . . ; ; : 10 10000 . . 5000 ( ) 15000 . 7500 . 10000 , 15000 7500 . , . ; 45 - 10 . 45 3/4 10 ( 3/2 ). : 10000(3/4)45 (3/2)10 1.38 !!!!!! 10. 10 http://mathhmagic.blogspot.gr/ 124 : , (1707-1783) " .." Franois Arago, Arago's Biographies of Scientific Men 1734-35, (1707-1783) , . . , : , . . 11. 11 http://mathhmagic.blogspot.gr/ 124 - - . , , . o , . , . 1) : . . . . .. . , ; x y, : x=100+(y-100)/10 x=200+(y-x-200)/10 x=300+(y-2x-300)/10 .. 100-(+100)/10 0 =900 y=8100, 9. 2) . , . , . , , . 24 . ; 12. 12 http://mathhmagic.blogspot.gr/ 124 ( Crux) x,y,z . x+y+z=s, s=72. , , s: x y z 2x-s 2y 2z 4x-2s 4y-s 4z 8x-4s=24 8y-2s=24 8z-s=24 s=72 , x=39,y=21,z=12 , Regula Caeci ( ). 3) , , 100 100 . 3 , 1 1/3 . . x,y,z , , . ,z . : +y+z=100 (7/2) x+ (4/3)y+ (1/2)z=100 21x+8y+3z=600 z y , y=60-18x/5 /5=t, . =5 y=60-18=42 , z=53 =10 y=60-36=24 , z=66 x=15 y=60-54=6 , z=79 x>15 y . 13. 13 http://mathhmagic.blogspot.gr/ 124 11. !! 15 24 . 8 . - : . ; - , ; 12. !!! : ; , . : , . . . ; 13. 8 . American Mathematical Monthly, 1956 14. !! . . : () . () . ; 14. 14 http://mathhmagic.blogspot.gr/ 124 4 . ( ) . . 3,4,5 . , . , , . ; 15. !!!! " Tsifuti Bank 500 !!! . 300 , 198 . 500 . : . - ;" 16. Fibonacci;; Komal. 3x3 , Fibonacci ; ( f1,f2, f1=f2=1). Fibonacci f0=0, f1=1,f2=1,.. fn+2=fn+fn+1, . : 0,1,1,2,3,5,8, 15. 15 http://mathhmagic.blogspot.gr/ 124 . . . . . Polya : , , . , . , : , . . , , , .( ) . . . . . , 0,1,2,3,4,..-1, . 0 -1 . -1 . , , . 16. 16 http://mathhmagic.blogspot.gr/ 124 (Dirichlet's Box Principle) (pigeonhole principle). Johann Peter Gustav Lejeune Dirichlet. , Dirichlet . , Dirichlet hm, Fourier, Laplace, Legendre Poisson. . 1855 Gauss, Gottingen. Johann Peter Gustav Lejeune Dirichlet(18051859) ( Dirichlet) , Recherches sur les forms quadratiques coefficients et indtermines complexes 1842. . . 3 2 . ; , , . 17. 17 http://mathhmagic.blogspot.gr/ 124 . : -1 , . . , . 15 ,15 /7 =2,1 . : +1 , +1 15=2* 7 +1 . , 1 , 2 ,1 ,., 1 ,2 ,3 ,,- +1 : 1>> >>> 10 . - >>> >>> 30 . - >>> >>> 15 . - >>> >>> 20 . 5 . ,,,, : , 20 , 20+10=30 , 20+10+30=60 , 20+10+30+15=75 20+30+60+75=185 . ; 5 6!=1x2x3x4x5x6=720 . - . . A1,A2,A t1,t2,t3,,t : 1:0 2:t1 A3: t1+t2 A4: t1+t2+ t3 . . 66. 66 http://mathhmagic.blogspot.gr/ 124 : t1+t2+ t3+.t-1 ( ) (-1)t1+(-2)t2+(-3)t3+..2t-2+t-1 (1) (-1) , (-2) . =5 : 10,15,20,20,30 (1) : (5-1)x10+(5-2)x15+(5-3)x20+(5-4)x20=4x10+3x15+2x20+1x20=40+45+40+20=145 ,,,, . 87. " " . ,, . ( ) . , ( ). , . . , , . 88. 100 . - - . 89. 100 , , , . 1 ( ) . . . . ; . . 67. 67 http://mathhmagic.blogspot.gr/ 124 90. . . : - , , ! ; 91. 25 , . . ( ). 92. 12,13,5 . ; 93 . , . , ( ) . : 12 . , . . ; 68. 68 http://mathhmagic.blogspot.gr/ 124 94.O 5 : , , , . 1. 5 : 30 30. 2. . 3. . 4. . 5. . 6. . 7. , 30. ; 95. !! . 10 10 10 20 . , , . 1 3 6 5 10:00 , ; (Crux magazine) 96. (!). . - ,,- . . . . . : " !! " ; 69. 69 http://mathhmagic.blogspot.gr/ 124 97. - , . , , , . : 1) . 2) . 3) . 4) . 5) 2800 . 6) . ; 98. 3 x,y,z. . : - . - 13. - : x2 ) PORK C CHOP = ( ). : C 3, 2 3, PORK . 3 9 PORK P CHOP = = 9 3 3 9 ORK HO = . =7 (3x9=27) 2 9 7 3 3 9 OR HO = ( ), : 118. 118 http://mathhmagic.blogspot.gr/ 124 9000 100 10 7 9000 300 30 27 100 10 7 300 30 27 100 10 300 30 20 7 30 2 X R H X X R H X X R H X X R H + + + = + + + + + = + + + = + + + = + 7+R 78+6=62 0,1,2. =0 7+R=2 =0 (=0). =1, 7+R=32 =4 R=4 (X=R). =2, 7+R=62 =8 R=6 . 9867 3 3289 = . .E. Dudeney : : - 5 =1 1 ( S,M) 4. - 4 S+M 9 , S+M 9 10 0 1. =1 =0. - 3 4 =9 =0. =0 S=9. 5 2 1 3 4 119. 119 http://mathhmagic.blogspot.gr/ 124 - 2 3 = =0. =0 S=9 , . =+1. - 1 2 +R 10 =+1 +1+R 10 R=9. S=9 1 2 R=8. - 1 2 D + E = 10 + Y. 0 1 , D + E 12. D 7 5. 7 N = E + 1. 5 6. - 6 D+E 12, D 7. N = E + 1 7 . =5 =6 - D+E 12 D=7 Y=2. S,E,N,D,M,O,R,Y 0,..,9 . 1814400 . : http://www.tkcs-collins.com/truman/alphamet/alphamet.shtml . , Bruce lee, , , . , . ... (invariant) . , , . Loren Problem-Solving Through Problems, (pursue parity) ( , ), (, , .)., . . 3 6 , ( ). , 53 . . ; 120. 120 http://mathhmagic.blogspot.gr/ 124 3 3.( ) 53 3 . : . / / / , , / / . ; . . 1,2,3,4,5,6. : =1+2+3+4+5+6=21. , . =1+2+3+4+5+6=21 1 2 3 4 5 6 1 2 3 4 5 6 121. 121 http://mathhmagic.blogspot.gr/ 124 1 2 , 6 2. =2+3+3+4+4+5=21. . 2 1 , 5 1., =3+3+3+4+4+4=21. . , , . . . 21, 6=21., 21 21., . 1 2 3 4 5 6 32 3 +2 544 -2 1 2 3 4 5 6 33 3 +1 444 -1 122. 122 http://mathhmagic.blogspot.gr/ 124 . . , : - , . - , . - , . , . ; ( ), 7 2 , . Terence Tao Solving Mathematical problems. 13 , 15 17 . , (.. ). , ; ,, ,, . (,,) . ( -1,-1,+2) (-1,+2,-1) (+2,-1,- 1). -,-,- (-=2,-=2 ,-=4) 3, -,-,- 3 ((-) Mod3=2,(-)mod3=2,(-)mod3=1,). , . . -=45,-=0,-=45 3 0 2. , . , . 23 32 ( ). . : . . . ( ) ., , , . : ; 0 ( ) 2 ( ). , ( ). , . ( ) . , : ; Obscurum per obscurius,ignotum per ignotius , Bruce Lee. 123. 123 http://mathhmagic.blogspot.gr/ 124 , , , . , . , . , . . , . Bruce Lee , , . . https://www.youtube.com/watch?v=5cpy1EPZHmI 124. 124 http://mathhmagic.blogspot.gr/ 124 125. 125 http://mathhmagic.blogspot.gr/ 124 1) 2) y . : 4 ( 1) 5 x : 4 4 ( ( 1) 1) 5 5 x : 4 4 4 4 4 ( ( { [( ( 1) 1)] 1} 1) 1) 1) 5 5 5 5 5 x : 1 4 4 4 4 4 ( ( ( { [( ( 1) 1)] 1} 1) 1) 1)) 5 5 5 5 5 5 y x= , , y . : 126. 126 http://mathhmagic.blogspot.gr/ 124 4 4 4 4 3 2 4 3 2 4 5 5 4 3 2 5 5 3 2 5 5 4 3 2 5 4 3 2 5 5 3 2 5 3 2 4 4 4 4 55 4 3 2 5 55 5 3 2 1 4 4 4 4 4 1 4 4 4 4 4 4 ( ( ( { [( ( 1) 1)] 1} 1) 1) 1)) ( ) 5 5 5 5 5 5 5 5 5 5 5 55 1 4 4 4 4 4 4 1 4 4 4 4 4 ( ) ( ( 1) ) 5 5 5 5 5 5 5 5 5 5 55 5 1 4 4 4 4 4 ( ( 1) ( ) 5 5 5 5 55 y x y x y x y x y x + + + = = = = = + + + 4 4 5 54 4 5 5 5 4 5 4 5 4 5 4 5 5 5 5 5 5 4 5 5 5 4 4 1 1 1 4 4 1 4 45 5( ( 1) ( ) ( ( 1) ( ) 4 15 5 5 5 5 51 5 5 1 4 4 4 4 ( ( 1) 4( 1)) ( 1) 4( 1)) 5 5 5 5 5 4 4 4 4 4 ( 1) 4 ( 1) 5 4 ( 1 5) 4 5 5 5 5 5 y x y x y x y x y x y x y x y = = = + = + = + = + = + 5 5 5 5 5 5 4 4 5 ( 4) 4 4 ( 4) ( 4) 4 5 5 4 x y x y x= + + = + + = 5 5 5 ( 4) 4 4 x y= + y y 5 5y k= : 5 5 5 (5 4) 4 4 x k= + 5 4k + 5 4 1024= . 5 4k + 5 4 5(4 ) 4 4(5 1)k + = + = + 4(5 1) 5 1 1024 256 + + = =51 =204 =3121. 3) ,+1,+2 3: +(+1)+(+2)+3=3(++1) 67 : 201(++1) > > > = > > ( ) =49 (2) 99 99 100 100 = = 99 100 1 , =99 =100. (,,)=(99,100,49). 8) : : x: y: (x,y ) : x+y : y+x 1 1 ( ) ( ) 1 ( ) ( ) 1 ( )( ) 1 ax y ay x ax y ay x a x y y x a x y x y a x y + = + + + = + = = = , , ,a x y x>y 1x y = 1 1a = = + . 1 . 9) . , 10000000 3162...KISS < 1,2,3 , =3 1I . 3100,3011,3022,3044,3055,3066,3077,3088,3099,3122,3144,3155, , ( PASSION ). ( PASSION ). 2 3. S 2,3,4,7,9 PASSION S ( 8 88 88 7744 = ). 132. 132 http://mathhmagic.blogspot.gr/ 124 2 SS 2 ISS , : S ON I K KISS 2 84 - 3 89 0 1 2 1033 2033 2 1 2 1233 2233 4 1 2 1433 2433 6 1 2 1633 2633 4 36 - 7 29 - 9 01 6 1 2 1699 2699 KISS 2033 PASSSION 4133089. 4133089 2033= 10) 5 . 3,6 9 , 3. . 3. 6x3+1x5+4x6+1x9=56 , , 28 . 28 3 , , 5 . 11) , . . 4 -24 4 24 15 -23 8 133. 133 http://mathhmagic.blogspot.gr/ 124 ( : ) : +=20 (1) +=15 (2) .(1),(2) =+5. , . : ++4=-24 +=-28 (3) 28 1 1 28 1 29 a a a + = + + + + 0 , 29 . 29. 12) , , 2 . 2 3y+2. 2 =3y +2 (1) ,y (1). : ) 3. ) 3 1. ) 3 2. - : ) =3( ) , 2 =92 =3y+2 3y+2 3. ) =3+1 , 2 =92 +6+1=3y+2. 92 +6=3y+1 3(32 +2)=3y+1 , 3y+1 3. ) =3+2 , 2 =92 +12+4=3y+2. 92 +12=3y-2 3(32 +4)=3y-2 , 3y-2 3. (1) . 13) x 8 ( x 1) : = [x(x + 7)][(x + 1)(x + 6)][x + 2)(x + 5)][(x + 3)(x + 4)] = = (x2 + 7x)(x2 + 7x + 6)(x2 + 7x + 10)(x2 + 7x + 12). x2 + 7x + 6 = y : = (y 6)(y)(y+ 4)(y + 6) = (y2 36)(y2 + 4y) = y4 + 4y(y+ 3)(y 12). y = x2 + 7x + 6 x 1 y 12 >0. (y 14) 134. 134 http://mathhmagic.blogspot.gr/ 124 > y4 . (1) , = y4 + 4y3 36y2 144y (y + 1)4 = y4 + 4y3 + 6y2 + 4y + 1. < (y+ 1)4 (2) (1),(2) : y4 < < (y + 1)4 y . 14) () ,, () . () . . 3 .a 4 1/3 . ,, 1/3 . , 1/3 . 3 !! 15) 300 198 , . 300 198 3 500 . 498 . ; 300 198 300 98 . +198+198-300+198-300 92 . 6 . 2 498 . 135. 135 http://mathhmagic.blogspot.gr/ 124 16) . , : : b+e+h=d+e+f b+h=d+f (1) d b,h,d,f . 0d ( d=0 a,c,g,i) .Y b h d b (1) : f h . d b h f . 0d (f1=f2=1 ) d b h f < . b,h Fibonacci f b h f+ (*) . b h f f d b h f d+ < + + < + (1) . 3x3 Fibonacci. (*) b,h f .) 17) . 1,2,3,4,, 9,10 5 10 ( 4 6 ). , ( ) 6 4 6-. . a b c d e f g h i (-4)+(10-)= 6 (6-)+(-4)=2 (6-)+(12+)=18 136. 136 http://mathhmagic.blogspot.gr/ 124 3. 6 , 2 -4. , 10 6-, , 8 12+.( ) , 7 14, : (12+)+=14 , =1 18) . 1 , 2 . 1 , 2 . 1 , 2 . 1 2 1 3 4 6 8 5 4 6 8 10 4 6 8 2 4 2 7 9 6 2 7 9 8 2 7 9 2 2 3 5 10 7 3 5 10 9 3 5 10 5 9 ( ) . 19) 200- . : 137. 137 http://mathhmagic.blogspot.gr/ 124 ( ) ( ). 2. , ( ) ( ). 200- . 150+50+(200-130)=270. : 2+200-=270 =70 70 . 20) : ( ). , : : 5 7 2 2 A A A+ = : 2 3B B B+ = : 6 7 + = : 6 7 + = , : 7 7 2 22 7 (1) 3 7 2 3 7 3 7 7 3 A A A A = = = = = = = = . 7 19 2 3 3 + + + = + + + = , 7 . 19 7 3 > , 19 2 7 3 3 = . ,, (1) . 138. 138 http://mathhmagic.blogspot.gr/ 124 21) : 8 0 0 8 0 0 3 5 0 5 0 3 3 2 3 5 3 0 6 2 0 2 3 3 6 0 2 2 5 1 1 5 2 7 0 1 1 4 3 7 1 0 4 4 0 4 1 3 4 4 0 22) 9 9 9 9 9 . WolphramAlpha 9 : 1 2 3 4 5 6 7 8 9 10 11 12 19 20 9 09 9 81 9 729 9 6561 9 59049 9 531441 9 4782969 9 43046721 9 387420489 9 3486784401 9 31381059609 9 ....81 . . 9 ...89 9 ....01 = = = = = = = = = = = = = = 139. 139 http://mathhmagic.blogspot.gr/ 124 ( .) 9 9 987420489 9 9 987420489 9 9= 89. 369 500 . 23) . x ( 1x ) , 1 x x x : 2 2 ( 1) 1 1 1 1 1 1 x x x x x x x x x x x x x x x + = + = + = 2 1 1 x x x x x = 24) () , () () ( ) , ( ) : , , , , , , , , , , 6 o . , , . 25) 20 . : . , , . 3 ( x 1 2x 2 , x+2x=3 x ) . 3 , 38 . 3 2 . 20 . 15 18 16 19 31 . 140. 140 http://mathhmagic.blogspot.gr/ 124 26) , , , : , , , 9 9 . 2+2=9 (1) , 2+2=2(+) 9 . 0 9: 0 9 0 2 18 0 2 18 ( ) 0 9 0 2 18 0 2 18 0 2( ) 36 0 18 0 9 18 0 2 a a a a a + + + =2. (1) : +=9 . . 8 9 4 8 . 4 4. 4 (00,44,88) 4 9 . :9900, 5544, 1188, 3636, 6336, 2772, 7272. 9900 5 00. 5544 8 1,2,3,4,6,7,8,9. 5544. . 27) i . i+(12-i)=12. 28) . , : - +10 : (-)-(+10)=--11 (1) , : +10 - : +10-(-)=-+11 (2) (1) ,(2) 11 , -=1 11 1 . - 141. 141 http://mathhmagic.blogspot.gr/ 124 0 =. . 29) : ,: . (), . 1 . 1. . 1 2 3 10 9 7 4 6 5 8 1 2 3 10 9 7 4 6 5 8 142. 142 http://mathhmagic.blogspot.gr/ 124 4 5 9. 5 . 9. 7( ) . 10 4, 4 10. 8 5. 3 6. . . 1 2 3 10 9 7 4 6 5 8 1 2 3 10 9 7 4 6 5 8 143. 143 http://mathhmagic.blogspot.gr/ 124 30) 4 5 10 2 = . : ( ) ( 4 ). 3 2.5+2++1.5+1+0.5=7.51) x y 3193 = . 3193 : 3193 31 103= , 103 50> x=31, 19 . 55) ,, . ( ) . , . ( ) ( ) 160. 160 http://mathhmagic.blogspot.gr/ 124 . 56) 1 2 62, ,.... 62 - . 3 5 = 62 2 = . i 1,2,...,59= : i 1 i i 2+ + = + i 2 i 1 i 3+ + + = + i i 3+ = (1) (1) i i 3+ i 3 i 6+ + = (2) i 1,2,...,56= . (1) ,(2) : i i 6+ = (3).: (3): 26 32 38 44 50 56 62 2 = = = = = = = 26 2 = . (2): 3 6 9 12 15 18 21 245 = = = = = = = = 24 245 5= = . 25 0 25 24 26 3 C = + = . 7 0 5 2 4 6 3 8 1 27 2 0 25 22 24 26 23 2 8 21 128 1 32 4 16 64 8 256 2 161. 161 http://mathhmagic.blogspot.gr/ 124 57) 1. =1. 1 G G=9 T=0. ,+=10 ( 1 +). 9( G=9). =8 1 +. G=9,H=1,R=2. R=2 ( ) 0+2, L. L 2, ( L+I) L=3. L+I (3+;) (1) . 8 ( =8). , =7 1 +. GOERING 6. =4. 1 7 03 6 2 9 8 6 27 4 9 1 0 0 3 31 1 1 + 58) 1 16 77 . i i . : 1 2 771 ... 132 < < < 21 1 2 7722 21 21 ... 21 153 + < + < < + 154 1 2 77 1 2 77, ,.., , 21, 21,.., 21 + + + i j i j 21 = + 21 j 1, j 2,...,i+ + . 59) ,, , . -2 . : ++=-2 : 2+= =-2 . 60) . - . 3 3 4 4 . 162. 162 http://mathhmagic.blogspot.gr/ 124 : 2 5 4 ( 4,5 4). 4 , . 2. 2 4 ( 4 5 2 4 2 4 ,). 4 3 5. 1,2,3,4,5 4 . 61) 17. . . . : : 3 ,7,11,15,19,23 . : 5,13,21. : 9,25. 9,25 1 17. 171 1 179 25 1 9 175 13 21 259 91 3 5 7 11 13 15 17 19 21 23 25 171 163. 163 http://mathhmagic.blogspot.gr/ 124 17 . , . 9 . . 62. , 150+70=1110 15+7=111 (1). (10 111 15 3 7 3 3. =3 , . (1) 15+7(3)=111 15+21=111 5+7=37 (2) . (2) , : 0 = + + + + = (1) : x,y 0 2xy 2xz 2yz xyz 2xz 2yz xyz 2xy 2xz 2yz xy(z 2) 0 2xz 2yz xy(z 2) xy(z 2) 0 z 2 0 z 2 (3) > + + = + = + = < + = > > > (1) ,(2) : 2 z 6< z {3,4,5,6} . 2xz 2yz xy(z 2)+ = xy z {3,4,5,6} 2xz 2yz 2z 2z 4z 4z 4z 2xz 2yz xy(z 2) xy xy (x y) (x x) x xy x y (4) z 2 z 2 z 2 z 2 z 2 z 2 + + = = = + + = (1) : 2yz 2(xy xz yz) xyz 2xy 2xz 2yz xyz 2xy 2xz xyz 2yz x(2y 2z yz) 2yz x yz 2y 2z + + + = + + = + = + = = 0 ,: z {3,4,5,6} 2z yz 2y 2z 0 yz 2y 2z y(z 2) 2z y z 2 > > > > (5) (4) (5) 2z 4z y z 2 z 2 < (6) z 3= y (6) : 7,8,9,10,11,12. 11 x. z 3= (42,7,3),(24,8,3),(18,9,3),(15,10,3) (12,12,3). 165. 165 http://mathhmagic.blogspot.gr/ 124 z 4= y 5,6,7,8 , 7 : (20,5,4),(12,6,4),(18,8,4). z 5= (10,5,5) z 6= (6,6,6). 10 : (42,7,3),(24,8,3),(18,9,3),(15,10,3),(12,12,3), (20,5,4),(12,6,4),(18,8,4), (10,5,5), (6,6,6). 68. ( x ) ( ) ( 1) 1 2 .. 2 ( 1) 2 ( 1) (2 1) x ( 1) ( 2) ... ( ) 1 2 3.. 2 2 2 + + + += + + + + + = + + + + + + = + + + + = + = = 2++1 . 2++1 . x . , ; , . : x pq= q(q 3 ) p . 2p q ( ). : 2p q< , q 2p 1 b ,(b 0) 2 = . ( q ) ( ) ( ) ( ) ( ) 2p (2p 1) 2p 1 q 2p 1 2p 1 q b 1 b 2 .... b 2p 2pb 1 2 ... 2p 2pb 2p(b ) 2p( )2p( ) pq x 2 2 2 2 2 + + + + + + + + + = + + + + = + = + = + = = 2p q> , q 1 b p ,(b 0) 2 + = ( q ) ( ) ( ) ( ) ( ) q (q 1) q 1 q 1 q 1 b 1 b 2 .... b q qb 1 2 ... q qb q(b ) q(p ) pq x 2 2 2 2 + + + + + + + + + + = + + + + = + = + = + = = . . 2 166. 166 http://mathhmagic.blogspot.gr/ 124 2 , 2048. 2048 . 69. y . .( ,x,y ) x y x y 20 + + ,: x y x y x y x x y y x( ) y x y 20 5 20 5 5 20 5 5 5 4 4 + = + = + = = + (1) x x y y x x( ) y y ( ) ( ) 1 4 4 20 5 4 4 1 4 20 5 9 20 5 4 5 4 9 > > = = > > > > (3) (2) ,(3): 20 4 3 9 < < = . 3= (1) : x y5 12= (4) 5 x+y : y x y x y x y y y y y y y y y 12 5 12 5 12 140 210 28 42 28 42 140 17 210 9 10 11 12 5 17 17 = = < + < < + < < < < < = = = = ( 28 4 42 6 ). (4): y x . . y x . y x . . y x . . 12 9 9 21 6 5 12 10 10 24 5 12 11 11 26 4 5 12 12 12 28 8 5 = = = = = = = = = = = = y=10 x=24. 24 . 70. . 200 300 . , . 249 249=83x3 . 83 3 3 83 . . 200 300 . 167. 167 http://mathhmagic.blogspot.gr/ 124 17 17^2=289. 17 17 . 71 1A = 2A = ( , ) ,,, , 0,1,2,3,,9 : ( ) ( ) ( ) ( ) ( )( ) ( )( )1 2A A 10 10 10 10 100 10 10 100 10 10 (1) = = + + = + + + + + = + + + = (1) : 12 42 21 24 504 12 63 21 36 756 12 84 21 48 1008 13 62 31 26 806 13 93 31 39 1209 14 82 41 28 1148 23 64 32 46 1472 23 96 32 69 2208 24 63 42 36 1512 24 84 42 48 2016 26 93 62 39 2418 34 86 43 68 2924 36 84 63 48 = = = = = = = = = = = = = = = = = = = = = = = = = 3024 46 96 64 69 4416 = = = 3000 4000 (36,84) (63,48) 63 84. , 84. 72. ( Microsoft,Google) Microsoft. (2000 , ) . 57,3% . . . 100 58 . Google. . ,,, . , ( ) . , . . 168. 168 http://mathhmagic.blogspot.gr/ 124 . . 73 100 , 10 90 . ( 10 ) x 10-x . (90 ) 90-x x . 90-x x . . 74. t 3( , , ) t=40+x , ( ). 60 , 20=80-60 . , 60=120-60 . ,t=40+x=40+(20+60+60)=180, 3 ( , ) 200-t=20. : 20/200=0.1 10% . 75. - x ( ) . - y ( A) . - z ( B) A . 60 2040 20 60 169. 169 http://mathhmagic.blogspot.gr/ 124 y z ( ) yz . : x+yz=33 (1) y+xz=23 (2) (1) ,(2) : (x+y)(z+1)=56 (3) (1) ,(2) : (y-x)(z-1)=10 (4) O z . (4) z-1 10. (3) z+1 56. z=3 z=6. z=3 x,y . z=6 x=3,y=5. :6+3*5=21. 76. . x ( ) . . , . : 51+49=100 5 ( 5) 49+5=54 51 ( 3) 51+5=56 49 ( 7) , . 105 . . 77. 1+2+3+4++50=1275 1271. . 25 25 2446. 2-1 ,2 4-1. 24 : 170. 170 http://mathhmagic.blogspot.gr/ 124 (41-1)+ (42-1)+ (43-1)+. +(424-1)=4(1+ 2+ 3+. +24)-24 4, 2446. ( 4.) 78. . 8 8. 6 6 . 8=6 (1) ; 8 . 8 , . , 4 . , 6/2=3. 8+5=4+3 (2). (1) ,(2) : =15 =20. 35 . 79. . (1) ,, . - . . . , (2) . . 80. , . . , . . 81 . 39 , 40 . ; 1 , , . , . . . . , .; . . . . ., , . . , , . , . .. . . . . , , 171. 171 http://mathhmagic.blogspot.gr/ 124 . 40 1 2 , . 40 . 82. = = = , , = = = . 3 3 2 2 3 3 3 3 3 3 1 = = = = = = = = 83. : : : : : : ) . ii) . 84. 3>35 : 12 ..20 21 22 23.. 48 49 50 .. 7>=43 : 7 8 9 10 11 12 ..20 21 22 23.. 48 49 50 .. 2==21 : 22 23.. 48 49 50 .. 5>=51 : 11 12 ..20 21 22 23.. 48 49 50 .. =11 1 4 . 85. . . . 172. 172 http://mathhmagic.blogspot.gr/ 124 . . , . 86. Hide Answer 87. . . ., . , , . 88. , - . 80 80 . , . ; , ; . , - -1 1 . - -1 , +(-1) . - +1 . - -2 +(-1)+(-2) , . 173. 173 http://mathhmagic.blogspot.gr/ 124 +(-1)+(-2)+(-3)++3+2+1 , : +(-1)+(-2)+(-3)++3+2+1>=100 ,(-1),(-2),(-3),,3,2,1 1 1, : +(-1)+(-2)+(-3)++3+2+1=(+1)/2 : (+1)/2>=100. , , Wolfram alpha : >=14. 14 . 14,27,39,50,60,69,77,84,90,95,100 . . 14 89. 50% . 50 . . 2,4,6,8,10,..,100 . - - 50 100 , 50 50% 25. 75% . ;; 174. 174 http://mathhmagic.blogspot.gr/ 124 50%. . : 1 98 , . 2 97 . 3 96 . 4 95 . . . . .... . ... , . ( ) ( ) 50%. . . . , . . . . ( ). . ( ) 20 13 20+13=33 . 99 . . 90. 2 x . 2 y , 2 x . : ( )( )2 2 y x 1111 y x y x 1111 11 101 = + = = 11 101 ., y x y x < + , : 175. 175 http://mathhmagic.blogspot.gr/ 124 1 y x 1 x 555 ... y x 1111 y 556 = = + = = 2 2 555 ,556 . 2 y x 11 x 45 ... y x 101 y 56 = = + = = 2 2 x 45 2025= = . 2025. 91. . , , . . ; , : 1. 2.. 3. 4. 5. , . . ; . , , 25 , , 15 5 . . : 1. 2. 3. 4. 5. 176. 176 http://mathhmagic.blogspot.gr/ 124 . . 14 . . 6 8 . 6 . ,, - , . 1 5 . ( ). . 7 3 25. 92. , , : 2 2 22 . : 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 4 1 1 1 2 0 ( 2 ) 0 2 0 4 4 4 8 4 4 0 8 4 4 8 8 ( 4) 16 4 0 ( 4) 4( 4) = + = = + + = + = + + = + + + + = + = + = + = + + = + = = 8 ( 4)( 4) 8 = -4 8 :2,3,5,6,8,12 , (6,8,10) , (5,12,13). 93. . , . . (Eureka) 1950 , Cambridge ( Archimedeans ) , Cambridge. Cedric A.B. Smith Blanchet Descartes. Smith . 12 : F,A,M, N,O,T ,L,I,C,K,E,D, 177. 177 http://mathhmagic.blogspot.gr/ 124 .. !!!!! , . : MA DO LIKE ME TO FIND FAKE COIN . , , . . . . , . . . : 1 2 3 . . . F . . . F . . . A . . . A . . . M . . . M . . . N . . . N . . . O . . . O . . . T . . . T . . . L . . . L . . . I . . . I 178. 178 http://mathhmagic.blogspot.gr/ 124 . . . C . . . C . . . K . . . K . . . E . . . E . . . D . . . D MA DO- LIKE ( ) (.) , ME TO FIND ( .) FAKE- COIN (.) . : . . . . , . 94. (1) ,(3) (4) . 30 - 30 . (7) . (2) ,(5) ,(6) , ( ). 30 . 95. y 10- 10-y . +6y 3(10-)+5(10-y)=80-3x-5y . , , . : +6y=80-3x-5y 4+11y=80 =(80-11y)/4 [0,10] (,y)=(9,4). x=9 y=4 33 33 . ; . 33 32 64 .: x+6y +80-3x-5y=80-2x+y 64 2x-y 16 ,y 10 : (8,0),(9,0),(9,1),(9,2),(10,0)(10,1),(10,2)(10,3),(10,4). 179. 179 http://mathhmagic.blogspot.gr/ 124 32 , : +6y 32 (1) 80-3x-5y 32 3x+5y 48 (2) (1) (10,4) (2) , 33 . , , 10:33. 96. . , , . , . . 97. : II III IV V VI 6) . ) III) . 1) 3) . III) V) . 3) ,4),5) 2) , 2800 3. II) IV). . ) VI). . ( ). . 180. 180 http://mathhmagic.blogspot.gr/ 124 . . 98. y . , ( ). : 4x : 4x 5y 6z : 15 13x y z+ + > !! 1, 2, 3x x x= = = - 3x = (3,4,5) ( y 5 y=5 3 11z - 10z = (1,2,10) z , , 10z . - 9z = , (1,3,9) , z . 9z , 8z . - 6z = (3,4,6) (2,5,6). 1 2 . . 6z . 5z = (3+4+5=1215 , r =16,17,18,19. 34 3 r r 3. r=18 . 112. . . , ,,. 1/8 , . . 1/8 , . . 189. 189 http://mathhmagic.blogspot.gr/ 124 , , . 1/8 2 . . 113. : 1111111=239x4649 239 4649 4649 239 114. a 10:00 , 10 . : ( )( )( ) (100 10 ) 100( ) 10( ) ( ) 100 10 100 100 10 10 100 10 100 89 10 100 10 100 89 10 0 100 ( 1) 10 ( 1) ( 1) 89 0 ( 1)(100 1 = + + + = + + + + + = + + + + + = + + + + + = + + = + 89 0 ) 89 0 ( 1) 89 0 ( 1) 89 0 1 X + = = = = X , 89 -1 1 ( 1,2). -1=1 =2 89x2=178 =356. ,178 10:00 356 12:00 2 . 115. . ,( ), 9 9 8 . . 10 . 116 . 10 513, : , , . ( ). , . , . . 500 512( 512 ) . 513 600 . 512 , 500 600, : 1,2,4,8,16,32,64,128,256,512,1024, 513 , ( 500 600) . 513 10 . , 512 9 , . 190. 190 http://mathhmagic.blogspot.gr/ 124 117. 9+31x2=71 , 60 : 71-60 =11 . 1, . 10,20,30 : 1000XXXXXXX 1,2 3 ( 31,32,33), 3 34 39 0 ( 40) 10001233330 ii) A 118. , 22 21 23, 50%. . . 1 2, 1 2.( 1). 2 3 . 2 1 2 3. ,+1 -1 -1 +1 . 1956 George Gamow Marvin Stern Puzzle Math. 191. 191 http://mathhmagic.blogspot.gr/ 124 119. 1 10 . ; . 5 . 1+2+3++10=55, 55+55=110 , 5( ) 22. 22. 1 2 2x21=42. 10 1 ( 10) . 4+5+6+7+8+9=39 + = = = = = = - , (1) + . , 2 =2. (1) : 2 (2 ) 120 2 120 ( 12)( 10) + = + + = + = , 10 1 = , 11 = . =11, =23 =2 . 138. 25 : 202. 202 http://mathhmagic.blogspot.gr/ 124 ++++++=25 (1) , , , : +=2(+) =-2 (2) : -1=++ (3) , , : ++=2(+) =+ (4) : ++++++=25 (1) =-2 (2) -1=++ (3) =+ (4) (1) (3) : ++++++ + -1 =25 + ++ 2++ +=26 (5) (2) -1 :- =-+2 (6) (4) -2 :-2 =-2-2 (7) (5) ,(6),(7): 2++ +- -2 =26-+2-2-2 4+ =26 =26-4 (8) (2) : =-2(26-4) =9-52 (9) , , , (8) 0 26-4 0 6 (10) (9) 0 9-52 0 52 9 5.77 ( ) 6 (11) (10) (11) =6. 139. : 203. 203 http://mathhmagic.blogspot.gr/ 124 : 25 : 16 : 20 : 25+16+20=225 (1) ++=12 =12-- (1): 25(12-- )+16+20=225 9+5=75 =(75-5)/9 =5(15-)/9 15- 9 =6 =5 ,=1 140. ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 2 3 1 2 3 1 3 2 1 3 21 2 3 : 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 3 21 3 2 3 2 2 2 3 1 3 2 6 21 2 2 2 3 2 2 2 2 2 A + + + + + + = = = = + + + + + + ++ + + + + = = = = + + H : 2 6 2 6 2 2 x y x y ++ = + = + (x,y) (2,6) ,(6,2) , . (1). ( ) ( ) 2 2 2 6 2 6 2 6 2 12 2 8 2 12 2 8 2 12 2 8 2( 12 ) (2) x y x y x y x y x y xy x y xy x y xy + = + + = + + + = + + + = + + + = + = (2) . ( ) 2 2( 12 ) 4(12 2 12 ) 4(12 4 3 )xy xy xy xy xy = + + = + + : 3xy 3xy 2 3xy n= (2) : ( )12 2 3 3 2 3xy n n = = . ( )2 3n 2n = . 12 0 12 12xy xy xy = = = . 204. 204 http://mathhmagic.blogspot.gr/ 124 (2) : 8 0 8x y x y+ = + = 8 8 12 12 (8 ) 2 , 6 x y y x xy x x x x + = = = = = = . : (2,6) ,(6,2). 141. 7 6 : 6 6 7 7 7 6 + = + 6 =6, , 7 =7 , , 6 6 6 7 6 6(1 7) 7 7 7 6 7 7(1 6) + + = = + + : (1 7) 1 1 7 1 6 7 6 (1 6) + = + = + = + , (1) : =6. =7. , 6 6 36, 7 7 49 = = = = +=36+49=85 . 142. , : 2 2 22 10 10 22x x x x + = = 0 2 0 10 22 0 ... 5 47 11 x x x x > + > (2) . 1 2 0 1 2 1 2 1 2 1 2... 10 10 .... 10 10 10 ....x a a a a a a a a a = = + + + = + + + ,: 1 1 2 0 1 2 1 1 2.... 9 10 10 .... 10a a a a a a a + + + 0 2 11 209 10 22 .... 13 2 2 x x x x x > + < (2) (1) ,(2) : 11 13 12 x x x < < = 205. 205 http://mathhmagic.blogspot.gr/ 124 143. (-) . .( ). 27 ( --.) . 27 , , . 144. 5 6 (1/12), y . ,y 1/12. : 1 1 1 1 1 12 12 12 12 12 x y x y x y x < < < < < + ,, (,y) o y=x+1/12, y=x-1/12. (11)2 =121 1-121/144=23/144. 15,97%. 206. 206 http://mathhmagic.blogspot.gr/ 124 145. . 5x5 . 13 12 . 25 , 13 . , pin , , : 131 112 132 98X p= + = + ,p : * 1000 9999 1000 132 98 9999 902 9901 902 132 9901 6.83 75.007 7 75(1) 132 132 p X p p p p p + 131 112 132 98 131 131 98 112 131( ) 14 (2)p p p p p + = + = + = . 0p p , (2) 207. 207 http://mathhmagic.blogspot.gr/ 124 0 14 131 145 131 14 14 131 14 131 117 p p p p p p p p p > = , (1) 7 75p . 0p = (2) : 14 0 14p p = = 132 14 98 1946X = + = pin 1946. http://www.slideshare.net/gdoubos/ss-33652426 146. . . . Thomas Alfa Edison. 147. , 2 , ( ) 39p q r + + = , 0

0 , >9, ) =5. .( 0 ) =0, 6 = =1 =6. =16,=61 =106 (61-16)/1=45 . a Y= , 0,1,4,5,6,9. : 100 11 (100 8) 3 4((25 2) 3 4 3 (1)a Y = = + = + + = + + = + ( 4 : 4) , 4 4+1 , . - ( ) 2 2 2 4 4 = = .(*) - ( ) 2 2 2 2 1 4 4 1 4( ) 1 4 1 + = + + = + + = + . 4 3a = + 4 4 1 + 0,1,4,5,6,9 4 = 0 = . - 0 = , 100a Y= Y . Y 2 2 2 2 {10 ,11 ,12 ,.....,31 }. 22 . - 4 = , 2 44 100 44Y Y = = + = . (*) 2 = 2 2 2 2 (2 ) 44 11 100 44 100 44 100 25 Y Y Y Y + = = = = : - 5 = ( ) 2 2 25 11 25 11 11 25 25 25 Y = = = , - 5 1 = + ( ) 2 2 25 1 11 25 10 1 11 2( 1) 25 25 5 Y + + + = = = + 214. 214 http://mathhmagic.blogspot.gr/ 124 {11,16,21,26,31} 5 . - 5 2 = + ( ) 2 25 2 11 20 7 25 5 Y + = = + , - 5 3 = + ( ) 2 25 3 11 30 2 25 5 Y + = = + , - 5 4 = + ( ) 2 2 25 4 11 8 1 3 1 25 5 5 Y + + + = = + = + + , 5 3, = + {13,18,23,28} 22+5+4=31 . 154. Polya, : x,y,z +y+z =9 1 x y z < < - - ( 9 ). 1+2+6=9 1+3+5=9 2+3+4=9 9 , 9 . ( ) 9, : 6 2 1 2 4 3 1 3 5 , 6 . 4. , . 215. 215 http://mathhmagic.blogspot.gr/ 124 , . , r , r . u r/u . 4 4u r/4u . r/u > r/4u, . , . r/4 r/2 r/2u , 2r /2u . . . r/4 , 180 ( ) , . r r 216. 216 http://mathhmagic.blogspot.gr/ 124 155. 11 . .To 2 , , . , . . , , ,.. 11 . , ( ). 16 , , . : 1,16,3,14,5,12,7,10,9,8,11,6,13,4,15,2 ( .) 156. 1 . 2 3 . 1 2 3 4 217. 217 http://mathhmagic.blogspot.gr/ 124 . 5 6 . 5 . 157. : 9 = , . 2 9 , =1 1 . (1) : 9 1 1 9 1 9 9 1 9 (1000 100 10 9) 9 1000 100 10 1 9000 900 90 81 9 1000 100 10 1 900 90 80 100 10 90 9 8 10 89 8 (2) = = + + + = + + + + + + = + + + + + = + + + = + + = ,,, (2) 0 = . 8 = . 1089 . 9 1089 9801 = . 158. , . , . , . . 2 . 2 , . 159. , (o 1000 ) , =100+10+ , 1a , 100 10 11 99 11 11 11(9 ) 11 11(9 ) 11 11(9 ) (1) a a a a a + + = + + + = + + + = + = + = (1) 11. : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 100 10 99 11 11 11 11 9 9 (2) 11 11 a a a a + + + + + = + + = + + = + + + + + + + + = + + + + = + + ,, : 1 9 1 9 1 9 0 9 0 9 9 0( ) 0 9 0 9 0 9 8 18, 11 + + + + + 0 11. 218. 218 http://mathhmagic.blogspot.gr/ 124 : 0 11 + = + = 1 0 + = (2) 2 2 2 9a + = + + ( ) 22 2 2 2 2 2 2 22 2 0 9 9 10 2 2 2 (3)9a a a + = + = + = + = + = + + + = + + + = + ++ + = + + + (3) . =0 ==5 =550. 2 2 2 2 2 2 2 8 10 2 10 10 8 2 4 8 2 2 2 10 8 ( 2 8) 2 4 8 10 8 2 6 0 3 0 ( 3) 0 0 3 1 2 + + + + + + + + = = = + + < + < < < < < = = =1 =2 . 2 2 2 2 2 2 2 2 2 11 11 11 .. 9 11 9 11 2 2 12 2 13 21 0(4)a a + = + = + + = + + + = + + + + = + + + + + = =0 (4) : 2 2 13 21 0 ( 3)(2 7) 0 + = = =3 =8. =803. >0 (4) : 2 2 2 2 2 2 2 2 2 12 13 21 0 2 2 2 12 20 1 13 2( 6 10) 1 13 + + + = + + + + = + + + + = (5) (5) . =1 1 11 10 + = = , , . >1 =3 , =0 , 5 , (4) : 2 2 2 2 2 2 2 12 13 21 0 2 2 12 21 2 13 + + + = + + + = + (6) 4 : 2 2 21 13 + > (6) : 2 2 0 2 22 2 12 2 2 12 2 12 2 18 2 12 18 6 9 3 9 2 18 > > + > + + < + < + < < =1 =2 219. 219 http://mathhmagic.blogspot.gr/ 124 =1 =2 ,. =550 ,=803 160. x . A 20 60 "" x - 20 60 , 16 72 , "" x - 16 72 . O , 60/(-20) = 72/ (-16) =40. 40. 161. 1 , S=1+2++=(1+ )/2 S = (1+ )/2 2S =( 1+ ) (1) (1) S 2 2S 2 ( 1+ ) 2, 2. 162. . 163. ) , 32 32 : 1+2+3++ =32 1447 47-44=3 43 =64 4 . http://www.wolframalpha.com/ : http://mathforum.org/k12/mathtips/beatcalc.html 250. 250 http://mathhmagic.blogspot.gr/ 124 . http://www.recordholders.org/en/events/worldcup/index.html , Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks, Arthur Benjamin Arithmetricks: 50 Easy Ways to Add, Subtract, Multiply, and Divide Without a Calculator, Edward H. Julius