Question

小学奥数，填数字

□□□+□□=□□+□□ ，把数字1~9填入上面的方框中，使等式成立，每个数字只能填一次，一共有多少种不同的填法？
答案是96种，我不知是怎么出来的？

Answer 1

96种是等号后两个不同数字交换位置算2种的。

按照等号后两数相加的进位，决定了等号前三位数的百位必是1。
1AB + CD = EF + GH能产生等价的4组：
1AB + CD = EF + GH
1AB + CD = GH + EF
1CD + AB = EF + GH
1CD + AB = GH + EF

等号后两数必要向百位进1。
等号前后个位数不进位的话，数字将不够用。
按以上限制，排列2到9使满足加法
AB + CD = EF + GH。即可

123 + 48 = 75 + 96
123 + 48 = 76 + 95
123 + 48 = 95 + 76
123 + 48 = 96 + 75
123 + 57 = 84 + 96
123 + 57 = 86 + 94
123 + 57 = 94 + 86
123 + 57 = 96 + 84
124 + 38 = 65 + 97
124 + 38 = 67 + 95
124 + 38 = 95 + 67
124 + 38 = 97 + 65
124 + 56 = 83 + 97
124 + 56 = 87 + 93
124 + 56 = 93 + 87
124 + 56 = 97 + 83
125 + 37 = 64 + 98
125 + 37 = 68 + 94
125 + 37 = 94 + 68
125 + 37 = 98 + 64
125 + 46 = 73 + 98
125 + 46 = 78 + 93
125 + 46 = 93 + 78
125 + 46 = 98 + 73
126 + 45 = 73 + 98
126 + 45 = 78 + 93
126 + 45 = 93 + 78
126 + 45 = 98 + 73
126 + 54 = 83 + 97
126 + 54 = 87 + 93
126 + 54 = 93 + 87
126 + 54 = 97 + 83
127 + 35 = 64 + 98
127 + 35 = 68 + 94
127 + 35 = 94 + 68
127 + 35 = 98 + 64
127 + 53 = 84 + 96
127 + 53 = 86 + 94
127 + 53 = 94 + 86
127 + 53 = 96 + 84
128 + 34 = 65 + 97
128 + 34 = 67 + 95
128 + 34 = 95 + 67
128 + 34 = 97 + 65
128 + 43 = 75 + 96
128 + 43 = 76 + 95
128 + 43 = 95 + 76
128 + 43 = 96 + 75
134 + 28 = 65 + 97
134 + 28 = 67 + 95
134 + 28 = 95 + 67
134 + 28 = 97 + 65
135 + 27 = 64 + 98
135 + 27 = 68 + 94
135 + 27 = 94 + 68
135 + 27 = 98 + 64
137 + 25 = 64 + 98
137 + 25 = 68 + 94
137 + 25 = 94 + 68
137 + 25 = 98 + 64
138 + 24 = 65 + 97
138 + 24 = 67 + 95
138 + 24 = 95 + 67
138 + 24 = 97 + 65
143 + 28 = 75 + 96
143 + 28 = 76 + 95
143 + 28 = 95 + 76
143 + 28 = 96 + 75
145 + 26 = 73 + 98
145 + 26 = 78 + 93
145 + 26 = 93 + 78
145 + 26 = 98 + 73
146 + 25 = 73 + 98
146 + 25 = 78 + 93
146 + 25 = 93 + 78
146 + 25 = 98 + 73
148 + 23 = 75 + 96
148 + 23 = 76 + 95
148 + 23 = 95 + 76
148 + 23 = 96 + 75
153 + 27 = 84 + 96
153 + 27 = 86 + 94
153 + 27 = 94 + 86
153 + 27 = 96 + 84
154 + 26 = 83 + 97
154 + 26 = 87 + 93
154 + 26 = 93 + 87
154 + 26 = 97 + 83
156 + 24 = 83 + 97
156 + 24 = 87 + 93
156 + 24 = 93 + 87
156 + 24 = 97 + 83
157 + 23 = 84 + 96
157 + 23 = 86 + 94
157 + 23 = 94 + 86
157 + 23 = 96 + 84

Answer 2

124+56=83+97
124+56=87+93
126+54=83+97
126+54=87+93
123 57 84 96
123 57 86 94
127 53 84 96
127 53 86 94

Answer 3

4

Answer 4

我知道，答案太多，这里太小写不出来。

Answer 5

123+57=94+86