Open Middle®
Directions: Use the digits 1 to 9, at most one time each, to make two true statements.
Hint
Answer
Number of Unique Solutions: 160
1: 1.2 + 3.5 = 4.7
2: 1.2 + 3.6 = 4.8
3: 1.2 + 3.7 = 4.9
4: 1.2 + 5.7 = 6.9
5: 1.2 + 7.4 = 8.6
6: 1.2 + 8.3 = 9.5
7: 1.3 + 4.9 = 6.2
8: 1.3 + 5.4 = 6.7
9: 1.3 + 5.9 = 7.2
10: 1.3 + 6.2 = 7.5
11: 1.4 + 6.5 = 7.9
12: 1.4 + 7.5 = 8.9
13: 1.4 + 7.8 = 9.2
14: 1.4 + 8.2 = 9.6
15: 1.5 + 2.8 = 4.3
16: 1.5 + 4.7 = 6.2
17: 1.5 + 4.8 = 6.3
18: 1.5 + 6.3 = 7.8
19: 1.5 + 6.7 = 8.2
20: 1.5 + 7.4 = 8.9
21: 1.5 + 7.8 = 9.3
22: 1.6 + 4.3 = 5.9
23: 1.6 + 5.8 = 7.4
24: 1.6 + 7.3 = 8.9
25: 1.6 + 7.8 = 9.4
26: 1.7 + 2.6 = 4.3
27: 1.7 + 2.9 = 4.6
28: 1.7 + 3.9 = 5.6
29: 1.7 + 4.2 = 5.9
30: 1.8 + 3.4 = 5.2
31: 1.8 + 4.7 = 6.5
32: 1.8 + 5.4 = 7.2
33: 1.8 + 7.5 = 9.3
34: 1.8 + 7.6 = 9.4
35: 1.9 + 2.7 = 4.6
36: 1.9 + 3.7 = 5.6
37: 1.9 + 6.3 = 8.2
38: 2.1 + 4.8 = 6.9
39: 2.1 + 5.3 = 7.4
40: 2.1 + 5.8 = 7.9
41: 2.1 + 6.3 = 8.4
42: 2.1 + 7.3 = 9.4
43: 2.1 + 7.4 = 9.5
44: 2.3 + 4.5 = 6.8
45: 2.3 + 6.4 = 8.7
46: 2.3 + 7.5 = 9.8
47: 2.4 + 1.5 = 3.9
48: 2.4 + 3.7 = 6.1
49: 2.4 + 5.7 = 8.1
50: 2.4 + 5.9 = 8.3
51: 2.4 + 6.1 = 8.5
52: 2.4 + 6.7 = 9.1
53: 2.5 + 1.4 = 3.9
54: 2.5 + 1.8 = 4.3
55: 2.5 + 4.8 = 7.3
56: 2.5 + 6.4 = 8.9
57: 2.5 + 6.8 = 9.3
58: 2.5 + 7.1 = 9.6
59: 2.5 + 7.3 = 9.8
60: 2.6 + 1.9 = 4.5
61: 2.6 + 4.5 = 7.1
62: 2.6 + 4.9 = 7.5
63: 2.6 + 5.7 = 8.3
64: 2.7 + 1.8 = 4.5
65: 2.7 + 3.4 = 6.1
66: 2.7 + 4.1 = 6.8
67: 2.7 + 5.6 = 8.3
68: 2.8 + 1.7 = 4.5
69: 2.8 + 3.1 = 5.9
70: 2.8 + 6.3 = 9.1
71: 2.8 + 6.5 = 9.3
72: 2.8 + 6.7 = 9.5
73: 2.9 + 1.6 = 4.5
74: 2.9 + 1.8 = 4.7
75: 2.9 + 3.5 = 6.4
76: 3.1 + 2.6 = 5.7
77: 3.1 + 4.5 = 7.6
78: 3.1 + 4.8 = 7.9
79: 3.1 + 6.7 = 9.8
80: 3.2 + 1.5 = 4.7
81: 3.2 + 1.7 = 4.9
82: 3.2 + 5.7 = 8.9
83: 3.4 + 5.7 = 9.1
84: 3.4 + 5.8 = 9.2
85: 3.5 + 4.1 = 7.6
86: 3.5 + 4.7 = 8.2
87: 3.6 + 1.8 = 5.4
88: 3.6 + 2.1 = 5.7
89: 3.6 + 4.9 = 8.5
90: 3.7 + 2.8 = 6.5
91: 3.7 + 5.2 = 8.9
92: 3.8 + 1.6 = 5.4
93: 3.8 + 2.1 = 5.9
94: 3.8 + 2.7 = 6.5
95: 3.8 + 5.4 = 9.2
96: 3.9 + 1.8 = 5.7
97: 3.9 + 2.5 = 6.4
98: 3.9 + 4.2 = 8.1
99: 3.9 + 4.7 = 8.6
100: 4.2 + 5.1 = 9.3
101: 4.3 + 2.5 = 6.8
102: 4.3 + 2.8 = 7.1
103: 4.5 + 1.7 = 6.2
104: 4.5 + 1.8 = 6.3
105: 4.5 + 2.8 = 7.3
106: 4.5 + 3.6 = 8.1
107: 4.5 + 3.7 = 8.2
108: 4.6 + 1.2 = 5.8
109: 4.6 + 2.5 = 7.1
110: 4.6 + 2.9 = 7.5
111: 4.6 + 3.2 = 7.8
112: 4.6 + 3.5 = 8.1
113: 4.6 + 3.9 = 8.5
114: 4.6 + 5.1 = 9.7
115: 4.7 + 1.2 = 5.9
116: 4.8 + 1.7 = 6.5
117: 4.8 + 2.3 = 7.1
118: 4.9 + 1.8 = 6.7
119: 5.1 + 3.6 = 8.7
120: 5.1 + 4.7 = 9.8
121: 5.2 + 1.7 = 6.9
122: 5.2 + 3.4 = 8.6
123: 5.2 + 4.1 = 9.3
124: 5.2 + 4.6 = 9.8
125: 5.3 + 2.6 = 7.9
126: 5.4 + 2.9 = 8.3
127: 5.6 + 3.1 = 8.7
128: 5.6 + 4.1 = 9.7
129: 5.6 + 4.2 = 9.8
130: 5.7 + 2.9 = 8.6
131: 5.7 + 4.1 = 9.8
132: 5.8 + 3.6 = 9.4
133: 6.1 + 3.4 = 9.5
134: 6.1 + 3.7 = 9.8
135: 6.2 + 3.5 = 9.7
136: 6.3 + 1.2 = 7.5
137: 6.3 + 2.4 = 8.7
138: 6.4 + 1.5 = 7.9
139: 6.4 + 1.9 = 8.3
140: 6.4 + 2.1 = 8.5
141: 6.4 + 3.1 = 9.5
142: 6.5 + 1.3 = 7.8
143: 6.5 + 1.7 = 8.2
144: 6.5 + 1.9 = 8.4
145: 6.5 + 2.4 = 8.9
146: 6.5 + 3.2 = 9.7
147: 6.8 + 2.3 = 9.1
148: 6.8 + 2.7 = 9.5
149: 6.9 + 1.4 = 8.3
150: 6.9 + 1.5 = 8.4
151: 7.1 + 2.4 = 9.5
152: 7.2 + 1.3 = 8.5
153: 7.2 + 1.4 = 8.6
154: 7.3 + 1.2 = 8.5
155: 7.5 + 2.1 = 9.6
156: 7.6 + 1.3 = 8.9
157: 8.2 + 1.3 = 9.5
158: 8.2 + 1.5 = 9.7
159: 8.4 + 1.2 = 9.6
160: 8.5 + 1.2 = 9.7
Source: Owen Kaplinsky
My fifth grade student was super proud she found 9 ways so far to complete this! She wanted me to post it here . Thanks!
I found out 20 ways to do this problem! Here are some ways:
4.8+3.1=7.9
3.2+4.7=7.9
7.1+2.3=9.4
1.5+2.4=3.9
8.3+1.4=9.7
I am in 5th grade and i found 3.4+5.2=8.6
I am in 4th grade and going to fifth grade after summer break. I found a lot of ways, on of them is 1.4+6.5=7.9
I am in Sixth grade and this is so good and fun to solve
The directions say to make “two true statements”, but it’s not possible to make two such statements each using 6 digits and have no digit be repeated. I guess the directions should say “to make *a* true statement”
The “solution”-section is missing a lot of solutions. There are 200 solutions, not just 160:
1.2 + 3.5 = 4.7
1.2 + 3.6 = 4.8
1.2 + 3.7 = 4.9
1.2 + 4.6 = 5.8
1.2 + 4.7 = 5.9
1.2 + 5.7 = 6.9
1.2 + 6.3 = 7.5
1.2 + 7.3 = 8.5
1.2 + 7.4 = 8.6
1.2 + 8.3 = 9.5
1.2 + 8.4 = 9.6
1.2 + 8.5 = 9.7
1.3 + 4.6 = 5.9
1.3 + 4.9 = 6.2
1.3 + 5.4 = 6.7
1.3 + 5.9 = 7.2
1.3 + 6.2 = 7.5
1.3 + 6.5 = 7.8
1.3 + 6.9 = 8.2
1.3 + 7.2 = 8.5
1.3 + 7.6 = 8.9
1.3 + 8.2 = 9.5
1.3 + 8.4 = 9.7
1.4 + 2.5 = 3.9
1.4 + 3.8 = 5.2
1.4 + 5.3 = 6.7
1.4 + 5.8 = 7.2
1.4 + 5.9 = 7.3
1.4 + 6.5 = 7.9
1.4 + 6.9 = 8.3
1.4 + 7.2 = 8.6
1.4 + 7.5 = 8.9
1.4 + 7.8 = 9.2
1.4 + 8.2 = 9.6
1.4 + 8.3 = 9.7
1.5 + 2.4 = 3.9
1.5 + 2.8 = 4.3
1.5 + 3.2 = 4.7
1.5 + 4.7 = 6.2
1.5 + 4.8 = 6.3
1.5 + 6.3 = 7.8
1.5 + 6.4 = 7.9
1.5 + 6.7 = 8.2
1.5 + 6.9 = 8.4
1.5 + 7.4 = 8.9
1.5 + 7.8 = 9.3
1.5 + 8.2 = 9.7
1.6 + 2.7 = 4.3
1.6 + 2.9 = 4.5
1.6 + 3.2 = 4.8
1.6 + 3.8 = 5.4
1.6 + 4.2 = 5.8
1.6 + 4.3 = 5.9
1.6 + 5.8 = 7.4
1.6 + 7.3 = 8.9
1.6 + 7.8 = 9.4
1.7 + 2.6 = 4.3
1.7 + 2.8 = 4.5
1.7 + 2.9 = 4.6
1.7 + 3.2 = 4.9
1.7 + 3.9 = 5.6
1.7 + 4.2 = 5.9
1.7 + 4.5 = 6.2
1.7 + 4.8 = 6.5
1.7 + 5.2 = 6.9
1.7 + 6.5 = 8.2
1.8 + 2.5 = 4.3
1.8 + 2.7 = 4.5
1.8 + 2.9 = 4.7
1.8 + 3.4 = 5.2
1.8 + 3.6 = 5.4
1.8 + 3.9 = 5.7
1.8 + 4.5 = 6.3
1.8 + 4.7 = 6.5
1.8 + 4.9 = 6.7
1.8 + 5.4 = 7.2
1.8 + 5.6 = 7.4
1.8 + 7.4 = 9.2
1.8 + 7.5 = 9.3
1.8 + 7.6 = 9.4
1.9 + 2.6 = 4.5
1.9 + 2.7 = 4.6
1.9 + 2.8 = 4.7
1.9 + 3.7 = 5.6
1.9 + 3.8 = 5.7
1.9 + 4.3 = 6.2
1.9 + 4.8 = 6.7
1.9 + 5.3 = 7.2
1.9 + 5.4 = 7.3
1.9 + 6.3 = 8.2
1.9 + 6.4 = 8.3
1.9 + 6.5 = 8.4
2.1 + 3.6 = 5.7
2.1 + 3.7 = 5.8
2.1 + 3.8 = 5.9
2.1 + 4.7 = 6.8
2.1 + 4.8 = 6.9
2.1 + 5.3 = 7.4
2.1 + 5.8 = 7.9
2.1 + 6.3 = 8.4
2.1 + 6.4 = 8.5
2.1 + 7.3 = 9.4
2.1 + 7.4 = 9.5
2.1 + 7.5 = 9.6
2.3 + 4.5 = 6.8
2.3 + 4.8 = 7.1
2.3 + 5.1 = 7.4
2.3 + 5.6 = 7.9
2.3 + 6.1 = 8.4
2.3 + 6.4 = 8.7
2.3 + 6.8 = 9.1
2.3 + 7.1 = 9.4
2.3 + 7.5 = 9.8
2.4 + 3.7 = 6.1
2.4 + 5.7 = 8.1
2.4 + 5.9 = 8.3
2.4 + 6.1 = 8.5
2.4 + 6.3 = 8.7
2.4 + 6.5 = 8.9
2.4 + 6.7 = 9.1
2.4 + 7.1 = 9.5
2.5 + 3.9 = 6.4
2.5 + 4.3 = 6.8
2.5 + 4.6 = 7.1
2.5 + 4.8 = 7.3
2.5 + 6.4 = 8.9
2.5 + 6.8 = 9.3
2.5 + 7.1 = 9.6
2.5 + 7.3 = 9.8
2.6 + 3.1 = 5.7
2.6 + 4.5 = 7.1
2.6 + 4.9 = 7.5
2.6 + 5.3 = 7.9
2.6 + 5.7 = 8.3
2.7 + 3.1 = 5.8
2.7 + 3.4 = 6.1
2.7 + 3.8 = 6.5
2.7 + 4.1 = 6.8
2.7 + 5.4 = 8.1
2.7 + 5.6 = 8.3
2.7 + 5.9 = 8.6
2.7 + 6.4 = 9.1
2.7 + 6.8 = 9.5
2.8 + 3.1 = 5.9
2.8 + 3.7 = 6.5
2.8 + 3.9 = 6.7
2.8 + 4.1 = 6.9
2.8 + 4.3 = 7.1
2.8 + 4.5 = 7.3
2.8 + 5.1 = 7.9
2.8 + 6.3 = 9.1
2.8 + 6.5 = 9.3
2.8 + 6.7 = 9.5
2.9 + 3.5 = 6.4
2.9 + 3.8 = 6.7
2.9 + 4.6 = 7.5
2.9 + 5.4 = 8.3
2.9 + 5.7 = 8.6
3.1 + 4.5 = 7.6
3.1 + 4.8 = 7.9
3.1 + 5.6 = 8.7
3.1 + 6.4 = 9.5
3.1 + 6.7 = 9.8
3.2 + 4.6 = 7.8
3.2 + 4.9 = 8.1
3.2 + 5.4 = 8.6
3.2 + 5.7 = 8.9
3.2 + 6.5 = 9.7
3.4 + 5.2 = 8.6
3.4 + 5.7 = 9.1
3.4 + 5.8 = 9.2
3.4 + 6.1 = 9.5
3.5 + 4.1 = 7.6
3.5 + 4.6 = 8.1
3.5 + 4.7 = 8.2
3.5 + 6.2 = 9.7
3.6 + 4.2 = 7.8
3.6 + 4.5 = 8.1
3.6 + 4.9 = 8.5
3.6 + 5.1 = 8.7
3.6 + 5.8 = 9.4
3.7 + 4.5 = 8.2
3.7 + 4.9 = 8.6
3.7 + 5.2 = 8.9
3.7 + 5.4 = 9.1
3.7 + 6.1 = 9.8
3.8 + 4.1 = 7.9
3.8 + 5.4 = 9.2
3.8 + 5.6 = 9.4
3.9 + 4.2 = 8.1
3.9 + 4.6 = 8.5
3.9 + 4.7 = 8.6
4.1 + 5.2 = 9.3
4.1 + 5.6 = 9.7
4.1 + 5.7 = 9.8
4.2 + 5.1 = 9.3
4.2 + 5.6 = 9.8
4.6 + 5.1 = 9.7
4.6 + 5.2 = 9.8
4.7 + 5.1 = 9.8
The “solution”-section for example has 2.5 + 1.8 = 4.3, 2.6 + 1.7 = 4.3, 2.8 + 1.5 = 4.3 but is missing 2.7 + 1.6 = 4.3.
Cool, 3.4 + 5.2 = 8.6 would work right?
I found so many ways, maybe 14