生於 1974

 

巧計零至百

1/9 * 7/4 * 0/4 * 2/3 = 0   1/9 * 7/4 * 0/4 = 0   1 * 9 - 7 - sqrt(4) = 0
1^974 + 0^423 = 1   1^974 - 0^4 = 1   -1 - 9 + 7 + 4 = 1
1 * 9 * 7 * 4 * 0 + (4 + 2)/3 = 2   1^(9 * 7) + 4^(0 * 4) = 2   1/(9 - 7) * 4 = 2
1 * 9/(7-4) + 0 * 4/(2-3) = 3   1 + 9 - 7 + 4 + 0 - 4 = 3   1 * 9/(7 - 4) = 3
-1 + sqrt(9) - 7 + sqrt(4) - 0 + sqrt(4) + 2 + 3 = 4   1 + 9/(7 - 4) + 0/4 = 4   1^(9 - 7) * 4 = 4
(1 + 9 + 7 + 4 + 0 + 4)/(2 + 3) = 5   (1 * 9 + 7 + 4)/(0 + 4) = 5   1 + (9 + 7)/4 = 5
(((1 * 9 + 7)/4 - 0) * 4 + 2)/3 = 6   (1 + 9)P(7 - 4)/(0! + 4)! = 6   sqrt(1 * 9 + 7) + sqrt(4) = 6
19 * 7 - 4 * 0 - 42 * 3 = 7   (1 + 9 - 7)! + (4 + 0)/4 = 7   1 + 9 - 7 + 4 = 7
1^(9 * 7) + 4 + 0^(4 * 2) + 3 = 8   1 * 9 + 7 - 4 - 0 - 4 = 8   1 * (9 - 7) * 4 = 8
(1 + 9 + 7 + 4 + 0 + 4 + 2)/3 = 9   1^9 + 7 + (4 + 0)/4 = 9   1 * sqrt(9) * (7 - 4) = 9
(1 + 9) * (7 - 4) - (0 + 4)*(2 + 3) = 10   1 * 9 - 7 + 4 + 0 + 4 = 10   (-1)^9 + 7 + 4 = 10
                     
(((1 + 9) * 7 + 4) * 0 + 4)* 2 + 3 = 11   1 + 9 - 7 + 4 - 0 + 4 = 11   1^9 * (7 + 4) = 11
1 + 9 - 7 + 4 - 0 + 4 - 2 + 3 = 12   1^9 * (7 - 4) * (0 + 4) = 12   1 * 9 + 7 - 4 = 12
1 * (9 - 7)^4 + 0^(4 - 2) - 3 = 13   1^9 + 7 + 4^0 + 4 = 13   1 + 9 + 7 - 4 = 13
1^97 + 4 + 0 + 4 + 2 + 3 = 14   1 + 9 + 74 * 0 + 4 = 14   1^9 * 7 * sqrt(4) = 14
1 * 9 + 7 * 4 * 0 * 4 + 2 * 3 = 15   -1 + 9 + 7 + 4 * 0 * 4 = 15   1 + sqrt(9) + 7 + 4 = 15
1 + 9 + 7 - 4 + 0 + 4 + 2 - 3 = 16   (1 + 9 * 7)/(4 + 0 * 4) = 16   (1 + 9 * 7)/4 = 16
1^9 + 7 + 4 + 0^4 + 2 + 3 = 17   1 * sqrt(9) + 7 * sqrt(4) + 0 * sqrt(4) = 17   1 + (9 - 7)^4 = 17
(1 + 9 - 7) * 4 + (0 + 4 - 2) * 3 = 18   1 * 9!/7! /(4 + 0 * 4) = 18   1 * 9!/7!/4 = 18
sqrt(1 * 9 + 7) * 4 + 0 + 4 + 2 - 3 = 19   1 + 9!/7!/(4 * 0 + 4) = 19   1 + 9 + 7 + sqrt(4) = 19
1 + (9 - 7) * (4 - 0) * (4 - 2) + 3 = 20   1 * 9 + 7 + 4 * 0 + 4 = 20   1 - 9 + 7 * 4 = 20
                     
(1 + 9)!/7!/4!/0! - 4 - 2 - 3 = 21   1 + 9 + 7 + 4 + 0^4 = 21   1 + 9 + 7 + 4 = 21
1!! + 9!!/7!! + 4!! + 0!! + 4!! - 2!! - 3!! = 22   (1 + 9) * (7 + 4)/(0! + 4) = 22   19 + 7 - 4 = 22
1 * sqrt(9) + 7 * sqrt(4) + 0 * sqrt(4) + 2 * 3 = 23   1 * 9 * (7 - 4) - 0 - 4 = 23   (-1)^97 + 4! = 23
1^9 + 7 + 4 + 0 + 4 + 2^3 = 24   (1 + 9 - 7) * (4 - 0 + 4) = 24   (1 + 9 - 7) * 4!! = 24
1 + 9 + 7 + 4 + (0 - 4) * (2 - 3) = 25   1 + 9 + 7 + 4 + 0 + 4 = 25   1 + ((9 + 7)/4)! = 25
1 * 9 + 7 + 4 * 0 + 4 + 2 * 3 = 26   (1 + 9)!/7!/4! - 0 - 4 = 26   -1 + 9 * (7 - 4) = 26
(1 + 9 - 7 + 4 * 0 * 4)^2 * 3 = 27   1 * 9 * (7 - 4) - 0 * 4 = 27   1 * sqrt(9)^(7 - 4) = 27
1 * (9 - 7) * 4 + (0 + 4) * (2 + 3) = 28   1 + sqrt(9)^(7 - 4) - 0 * 4 = 28   1^9 * 7 * 4 = 28
1 * 9 + 7 * 4 - 0^4 - 2^3 = 29   1^9 + 7 * 4 + 0^4 = 29   1^9 + 7 + 4 = 29
1 + 9 + 7 + 4 + 0 + 4 + 2 + 3 = 30   (1 + 9)C(7 - 4)/(0 + 4) = 30   (1 + 9)C7/4 = 30
                     
1 * 9 + 7 * 4 - 0 * 4 - 2 * 3 = 31   (1 + 9) * (7 - 4) + 0!^4 = 31   -1 + 9C7 - 4 = 31
(1 + 9)!/(7! * 4!) + (0 + 4)!/(2! * 3!) = 32   1 * (9 - 7) * 4 * (0 + 4) = 32   (1^9 + 7) * 4 = 32
1 * 9 * (7 - 4) + 0 * 4 + 2 * 3 = 33   1 - 9 - 7 + 4! + 0 + 4! = 33   1 * sqrt(9) * (7 + 4) = 33
(1 + 9) * (7 - 4) - (0 + 4) * (2 - 3) = 34   (1 + 9) * (7 - 4) + 0 + 4 = 34   (1 + 9 + 7) * sqrt(4) = 34
1 * 9 * 7 - 4 + 0 - 4 * 2 * 3 = 35   (1 * 9)C7 - (4 + 0)C4 = 35   1 + sqrt(9) + 7 + 4! = 35
1 + 9 + 7 + 4 + 0 + 4P2 + 3 = 36   1 * 9C7 * 4C(0 + 4) = 36   (1 + sqrt(9)) * (7 + sqrt(4)) = 36
1^9 + 7 * 4 - 0 * 4 + 2^3 = 37   1 * 9 + 7 * 4 + 0 * 4 = 37   1 + 9!!/7!! * 4 = 37
(1^9 + 7) * 4 + (0^4 + 2) * 3 = 38   1 + 9C7 + (4 + 0)/4 = 38   1 + 9 + 7 * 4 = 38
1^9 * 7 + 4 * 0 + 4 * 2^3 = 39   1 + 9 + 7 * 4 + 0!^4 = 39   -1 + 9C7 + 4 = 39
1 * (9 + 7 + 4) * (0 + 4 + 2)/3 = 40   (1 + sqrt(9)) * (7 - sqrt(4)) * (0 + sqrt(4)) = 40   1 + 9 * 7 - 4! = 40
                     
1! *  9 * 7 - 4! + 0! * 4 * 2 - 3! = 41   1 + 9C7 * 4C0 + 4 = 41   1 + 9C7 + 4 = 41
1 * 9 * 7 * 4/(0 * 4 + 2 * 3) = 42   1 + 9C7 + 4C0 + 4 = 42   1 * sqrt(9) * 7 * sqrt(4) = 42
1 * 9 + 7 * 4 + 0 * 4 + 2 * 3 = 43   1 + sqrt(9) * 7 * sqrt(4) + 0 * sqrt(4) = 43   1 + sqrt(9) * 7 * sqrt(4) = 43
1^9 * (7 + 4) * (0 - 4) * (2 - 3) = 44   1^9 * (7 + 4) * (0 + 4) = 44   (1 + sqrt(9) + 7) * 4 = 44
1 * 9C(7 - 4) - 0 - 42 + 3 = 45   (1 + sqrt(9)) * (7 - sqrt(4)) + 0! + 4! = 45   1 + 9 +7C4 = 45
(1 + 9 - 7)! + (4 + 0)! + 4! - 2^3 = 46   (-1 * 9 + 7) + sqrt(4) * (0 + 4)! = 46   -1 * sqrt(9) + 7^sqrt(4) = 46
1 + 9C7 + 4C0 + 4C2 + 3 = 47   1 * 9!/7! - 4C0 - 4! = 47   -1! + 9!/7! - 4! = 47
(19 - 7) * 4 + 0 * (4 - 23) = 48   ((1 + 9 - 7) * 4 - 0) * 4 = 48   (19 - 7) * 4 = 48
(1 + 9 - 7 + 4)^((0 + 4 + 2)/3) = 49   1 * 97 - 4! - 0 - 4! = 49   1 + (9 - 7) * 4! = 49
(1 + 9) * 7 - 4 * 0 - 4 * (2 + 3) = 50   -19 + 74 - 4 - (0 * 4)! = 50   1^sqrt(9) + 7^sqrt(4) = 50
                     
1 * 9 * (7 - 4 - 0) + 4 * 2 * 3 = 51   1! + (9 - 7)! + (4 - 0)! + 4! = 51   1 * sqrt(9) * (-7 + 4!) = 51
1 + (9 - 7) * (4 + 0) * (4 + 2) + 3 = 52   (1 + 9 + 7 - 4) * (0 + 4) = 52   (1 * sqrt(9)! + 7) * 4 = 52
1! + 9C7 + 4C0 + 4C2 + 3! = 53   1 * 9 + (7 + 4) * (0 + 4) = 53   (1 + sqrt(9)!) * 7 + 4 = 53
1! * 9!/7! - 4! + 0! * 4!/2! - 3! = 54   1 * 9 * 7 - 4 - 0! - 4 = 54   1 * 9 * (7 - 4)! = 54
(1 + 9) * (7 + 4) * (0 + 4)/2^3 = 55   19  + 7 + 4 + 0! + 4! = 55   1 + 9 * (7 - 4)! = 55
1 * 9 * 7 - 4 + 0 * 4 * 2 - 3 = 56   1 * 9C7 + 4 *  (0! + 4) = 56   (-1 + 9) * sqrt(7)^sqrt(4) = 56
19 + 7 + 4 + 0 + 4 + 23 = 57   (1 + 9) * 7 - 4!! - 0! - 4! = 57   (1 * 9 + 7!!)/sqrt(4) = 57
(1 + 9) * (7 + 4)/(0 + 4 - 2) + 3 = 58   (-1 + 9) * 7 + 4 + sqrt(0) - sqrt(4) = 58   1 * 9 + 7^sqrt(4) = 58
1 + 9 + (7 + 4) * (0 + 4) + 2 + 3 = 59   (1 + 9 + 7) * 4 - 0!! - 4!! = 59   1 + 9 +7^sqrt(4) = 59
1! + (9 - 7)! + 4! + 0! + 4! + 2! + 3! = 60   (1 + 9) * (7 - (4 + 0)/4) = 60   (1 + 9) * (7 - 4)! = 60
                     
1 + 9 + 74 - 0 * 4 - 23 = 61   1 * 9 * 7 - 4!!/(0 + 4) = 61   -1 + sqrt(9)! + 7 * 4!! = 61
(1 + 9 - 7)! + 4! + (0 + 4)! + 2! + 3! = 62   (1 + 9) * 7 - 4 - 0 - 4  = 62   (1 + 9) * 7 - 4!! = 62
1 + 97 + 4 + 0 - 42 + 3 = 63   (1 * 9 * 7 * 4)/(0 + 4) = 63   1 * 9 * sqrt(7)^sqrt(4) = 63
(1 * 9 + 7) * 4 + 0 * (4 + 2 * 3) = 64   1 * 9 * 7 + 4^(0 * 4) = 64   (1^9 + 7)^sqrt(4) = 64
(1 * 9 + 7) * (4 * 0 + 4) - 2 + 3 = 65   (1 - sqrt(9) + 7) * (4!! + 0! + 4) = 65   1 + (9  + 7) * 4 = 65
19 + 74 - 0 - 4 - 23 = 66   ((-1)^9 + 7) * (4!! - 0! + 4) = 66   1 - 9 + 74 = 66
19 + 7 + 4! + 0! + 4! -  2^3 = 67   1 * 9 * 7 + 4 * 0 + 4 = 67   1 * 9 * 7 + 4 = 67
(1 + 9 + 7) * 4 + 0 * (4 + 2 + 3) = 68   19 + 74 - 0! - 4! = 68   (1 + 9 +7) * 4 = 68
1 * 9 * 7 + 4 * 0 * 4 + 2 * 3 = 69   (1 + 9) * 7 - (4 + 0)/4 = 69   -sqrt(1)! - sqrt(9)! + 74 = 69
19 + 7 * 4 + 0 * 4 + 23 = 70   (1!! + 9!!/7!!) * (4 + 0! + 4)  = 70   (1 + 9) * sqrt(7)^sqrt(4) = 70
                     
(1 + 9 + 7) * (4 + 0 + 4)/2 + 3 = 71   1 * 9C7 * sqrt(4) - 0!^4 = 71   -1 + 9C7 * sqrt(4) = 71
1 * 9C7 * 4C0 * 4C2/3 = 72   1! + 9!/7! + 4! - 0! - 4! = 72   1!! * 9!!/7!! * 4!! = 72
(1 + 9 - 7)^4 - (0 + 4 - 2)^3 = 73   (1 * 9)!/7! + (4 + 0)!/4! = 73   1 + 9C7 * sqrt(4) = 73
19 + 74 - 0 - 4 + 23 = 74   1^9 * 74 - 0^4 = 74   (1 + 9) * 7 + 4 = 74
1 + 9 * 7 + 4 * 0 + 4 * 2 + 3 = 75   1^9 + 74 + 0^4 = 75   1^9 + 74 = 75
1 + 9C(7 - 4) - (0 + 4)C2 - 3 = 76   (1 + 9) * 7 + 4!/(0 + 4) = 76   1 * 9!/7! + 4 = 76
1 * 9 * 7 + 4 * 0 + 4 * 2 + 3 = 77   19 + 7C4 - 0! + 4! = 77   1 + 9!/7! + 4 = 77
(1 + 9 + 7 + 4) * (0 + 4) - 2 * 3 = 78   1^9 * 74 + 0 + 4 = 78   (1 + 9) * 7 + 4!! = 78
1 * 9!/7! + 4 + 0 * 4!/2! + 3 = 79   -1!! + 9!!/7!! * 4!!/0!! + 4!! = 79   -1 + (sqrt(9) + 7) * 4!! = 79
(1 + 9) * 7 + (4 + 0) * 4 - 2 * 3 = 80   (1 + 9) * (7 + 4/(0 + 4)) = 80   (1!! + 9!!/7!!) * 4!! = 80
                     
(1 + 9 - 7)^4 + 0^(4 - 2 + 3) = 81   (1 + 9 - 7)^(4 + 0 * 4) = 81   (1 + 9 - 7)^4 = 81
(1 + 9) * 7 + 4 + 0^4 + 2^3 = 82   (1 + 9) * 7 + 4!! + 0 + 4 = 82   1 + 9 * (7 + sqrt(4)) = 82
19 + 74 + 0! - 4 * 2 - 3 = 83    1 *  9 + 74 + 0 * 4 = 83   1 * 9 + 74 = 83
19 + 74 - 0 - 4 - 2 - 3 = 84   (1 + 9 + 7 + 4) * (0 + 4) = 84   1 * 9C(7 - 4) = 84
19 + 74 - 0^4 - 2^3 = 85   1 + 9C(7 - 4) + 0 * 4 = 85   1 + 9C(7 - 4) = 85
(1 + 9 + 7) * 4 + (0 + 4 + 2) * 3 = 86   (1 + 9) * 7 + (4 + 0) * 4 = 86   -1! + sqrt(9)! + 7!! - 4! = 86
1 * 9 * 7 + 4 * 0 + 4 * 2 * 3 = 87   1! * 9 + 74 + 0! * 4 = 87   1 * 9 * 7 + 4! = 87
(1 + 9) * (7 +4) * (4 + 0)/(2 + 3) = 88   1 + 9 + 74 + 0 + 4 = 88   (-1 + 9) * (7 + 4) = 88
(1 + 9 - 7)^4 + (0 + 4 - 2)^3 = 89   19 + 74 - 0 - 4 = 89   1 - 9 + 7!! - 4!! = 89
(1 + 9) * (7 + 4) - (0 + 4) * (2 + 3) = 90   1 * 9 * (7 + 4 - 0!^4) = 90   (sqrt(1) + 9) * (7 + sqrt(4)) = 90
                     
(1 + 9) * 7 + (4 + 0) * 4 + 2 + 3 = 91   1^9 * 7 * (4!! + 0! + 4) = 91   -1 - 9 + 7!! - 4 = 91
1 + 9 * 7 + 4 + 0 + 4 * 2 * 3 = 92   19 + 74 - 0!^4 = 92   -1 * 9 + 7!! - 4 = 92
1 + 97 + 40 - 42 - 3 = 93   19 + 74 + 0 * 4 = 93   19 + 74 = 93
1 + 9C(7 - 4) + (0 + 4)C2 + 3 = 94   1 + 97 -  4 - 0 * 4 = 94   1 + 97 - 4 = 94
1 + 9 + 74 + 0 + 4*2 + 3 = 95   -1 + 97 - 4/(0 + 4) = 95   1 * 97 - sqrt(4) = 95
19 + 74 + 0 + 4 + 2 - 3 = 96   (1 + 9)C7 - 4P(0 + 4) = 96   (1 + 9)C7 - 4! = 96
(1!! + 9!!/7!!) * (4!! + 0!!) + 4!! + 2!! - 3!! = 97   19 + 74 + 0 + 4 = 97   1^9 * 7!! - 4!! = 97
1^9 + 74 + 0^4 + 23 = 98   1 * (9 - 7)! + 4 * (0 + 4)! = 98   -1 + 9 * (7 + 4) = 98
19 + 74 + 0 * 4 + 2 * 3 = 99   -1 + 9!/7! + 4!/0! + 4 = 99   1 * 9 * (7 + 4) = 99
(1 + 9)^(7 + 4 - 0 - 4 - 2 - 3) = 100   (19 + 7) * 4 - 0 - 4 = 100   -1 + 97 + 4 = 100

註:算式除了使用了加 (+)、減 (-)、乘 (*)、除 (/)、冪 (^)、括號以外,還使用了下列運算符號:

平方根 sqrt(x):如 sqrt(9) = 3 ,sqrt(1*9+7) = sqrt(16) = 4。

階乘 n! = n*(n-1)*(n-2)*...:如 5! = 120,3! = 6,0! = 1。

雙階乘 n!! = n*(n-2)*(n-4)*...:如 7!! = 105,4!! = 8,0!! = 1。

組合 nCr = n!/[(n-r)!r!]:如 9C7 = 36,4C0 = 1。

排列 nPr = n!/(n-r)!:如 9C(7-4) = 504,4P(0+4) = 24。

拼合:即把相鄰的兩個或多個數字看成一個兩位數或多位數,如把 1974 看成一個 4 位數。

 

 

創作巧計的過程

我生於一九七四年四月二十三日。看罷前輩的大作,蠢蠢欲動,望在二零一一年生日前後,利用生日數字作個宣傳小玩意。我是規定了數字的順序,原因是我想突顯生日日期,這也比不定字序難一些。原來創作過程不過數天,說難亦可,說易亦是,但總算總結了一些心得,供諸同好。

 

開始之時,我先擬定一些看來較美觀或對稱的算式,如:

1 + 9 + 7 + 4 + 0 + 4 + 2 + 3 = 30

1 * 9 + 7 * 4 + 0 * 4 + 2 * 3 = 43

1! + 9C7 + 4C0 + 4C2 + 3! = 53

19 + 7 + 4 + 0 + 4 + 23 = 57

1 * 9!/7! + 4 + 0 * 4!/2! + 3 = 79

這樣,我完成了約三十道算式。

 

其實起初希望只用四則和方冪而已,但漸漸感到有些算式弄不成,才增加運算符號而已。這兒我想,若減少運算方式,我可能要多想數天了。

 

接下來,我得看看各個數字的大能如何,如 9,可以是 9,可以是 sqrt(9) = 3,可以是 sqrt(9)! = 6等,又看看兩個數字或三個數字的組合有哪些可能性,再拼合出未完成的算式來。

 

完成八個數字以後,再向六個數字和四個數字挑戰,由於數字數目少了,也相應難了。說來八個數字的算式在二零一零年十二月左右完成,以期翌年生日發佈。而六個數字和四個數字則在二零一一年四月左右構思,五月十七日完成。那夜,我尚餘四個數字中的四道算式 (69 、75、86 和 89) 未許攻克,往慕道班的路途上,偶爾清風送爽,便找到了一些數值,算式便來了。這樣,我除了感謝神,還可說什麼呢。我苦思了數日也想不到的,一下了通了。

 

說回來計算還得仰仗我出生的日子不俗,當中有很多大大小小不同的數字,增加了數值的多樣性。更重要的是當中有一個進可攻,退可守的零:零可把多餘的數字「吃掉」,又可在加法中「自動消失」,必要時更可以化身 0!=1,充當一角。零這回真的可以說句:「別少看我耶」。

 

參考文獻及網址

劉松基 "五秩進一" 自 數學中年漢的自述 , 香港 : 教育局課程發展處數學教育組 , p. 103-117 , 2010

 

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