Open Middle®
Directions: Use the digits, 0 through 9, without repeats, to complete the equation below:
Hint
Answer
Number of Unique Solutions: 274
1: 1.2 + 3.5 = 4.7
2: 1.2 + 3.6 = 4.8
3: 1.2 + 4.6 = 5.8
4: 1.2 + 5.7 = 6.9
5: 1.2 + 6.3 = 7.5
6: 1.2 + 7.4 = 8.6
7: 1.2 + 8.4 = 9.6
8: 1.3 + 7.2 = 8.5
9: 1.3 + 7.6 = 8.9
10: 1.3 + 8.2 = 9.5
11: 1.4 + 6.9 = 8.3
12: 1.4 + 7.5 = 8.9
13: 1.4 + 8.2 = 9.6
14: 1.5 + 2.8 = 4.3
15: 1.5 + 4.7 = 6.2
16: 1.5 + 4.8 = 6.3
17: 1.5 + 6.3 = 7.8
18: 1.5 + 6.4 = 7.9
19: 1.5 + 6.7 = 8.2
20: 1.5 + 6.9 = 8.4
21: 1.5 + 7.4 = 8.9
22: 1.5 + 7.8 = 9.3
23: 1.5 + 8.2 = 9.7
24: 1.6 + 3.8 = 5.4
25: 1.6 + 4.3 = 5.9
26: 1.6 + 7.8 = 9.4
27: 1.7 + 2.6 = 4.3
28: 1.7 + 3.9 = 5.6
29: 1.7 + 4.2 = 5.9
30: 1.7 + 4.8 = 6.5
31: 1.8 + 3.4 = 5.2
32: 1.8 + 4.5 = 6.3
33: 1.8 + 4.9 = 6.7
34: 1.9 + 6.5 = 8.4
35: 2.1 + 3.6 = 5.7
36: 2.1 + 3.8 = 5.9
37: 2.1 + 5.3 = 7.4
38: 2.1 + 5.8 = 7.9
39: 2.1 + 6.3 = 8.4
40: 2.1 + 7.3 = 9.4
41: 2.1 + 7.5 = 9.6
42: 2.3 + 6.8 = 9.1
43: 2.3 + 7.5 = 9.8
44: 2.3 + 8.4 = 10.7
45: 2.4 + 1.5 = 3.9
46: 2.4 + 3.7 = 6.1
47: 2.4 + 5.9 = 8.3
48: 2.4 + 6.3 = 8.7
49: 2.4 + 6.5 = 8.9
50: 2.4 + 7.1 = 9.5
51: 2.4 + 8.5 = 10.9
52: 2.5 + 1.4 = 3.9
53: 2.5 + 1.8 = 4.3
54: 2.5 + 6.4 = 8.9
55: 2.5 + 7.1 = 9.6
56: 2.5 + 7.8 = 10.3
57: 2.6 + 1.9 = 4.5
58: 2.6 + 3.1 = 5.7
59: 2.6 + 4.9 = 7.5
60: 2.6 + 5.3 = 7.9
61: 2.7 + 1.8 = 4.5
62: 2.7 + 1.9 = 4.6
63: 2.7 + 3.8 = 6.5
64: 2.7 + 6.8 = 9.5
65: 2.8 + 1.7 = 4.5
66: 2.8 + 3.1 = 5.9
67: 2.8 + 4.5 = 7.3
68: 2.8 + 6.3 = 9.1
69: 2.8 + 6.5 = 9.3
70: 2.9 + 1.6 = 4.5
71: 2.9 + 1.7 = 4.6
72: 2.9 + 1.8 = 4.7
73: 2.9 + 3.5 = 6.4
74: 2.9 + 5.4 = 8.3
75: 2.9 + 5.7 = 8.6
76: 2.9 + 7.4 = 10.3
77: 3.1 + 4.5 = 7.6
78: 3.1 + 4.8 = 7.9
79: 3.2 + 1.5 = 4.7
80: 3.2 + 1.7 = 4.9
81: 3.2 + 5.7 = 8.9
82: 3.4 + 2.7 = 6.1
83: 3.4 + 6.8 = 10.2
84: 3.4 + 8.6 = 12.0
85: 3.5 + 4.1 = 7.6
86: 3.5 + 6.2 = 9.7
87: 3.5 + 6.7 = 10.2
88: 3.5 + 8.9 = 12.4
89: 3.6 + 1.8 = 5.4
90: 3.6 + 4.5 = 8.1
91: 3.6 + 4.9 = 8.5
92: 3.6 + 7.2 = 10.8
93: 3.6 + 8.4 = 12.0
94: 3.7 + 1.2 = 4.9
95: 3.7 + 1.9 = 5.6
96: 3.7 + 2.8 = 6.5
97: 3.7 + 5.2 = 8.9
98: 3.7 + 6.1 = 9.8
99: 3.7 + 6.5 = 10.2
100: 3.7 + 6.8 = 10.5
101: 3.8 + 5.4 = 9.2
102: 3.8 + 6.4 = 10.2
103: 3.8 + 6.7 = 10.5
104: 3.8 + 6.9 = 10.7
105: 3.9 + 1.8 = 5.7
106: 3.9 + 2.5 = 6.4
107: 3.9 + 4.2 = 8.1
108: 3.9 + 6.5 = 10.4
109: 3.9 + 6.8 = 10.7
110: 4.1 + 2.7 = 6.8
111: 4.1 + 5.2 = 9.3
112: 4.1 + 5.7 = 9.8
113: 4.2 + 5.1 = 9.3
114: 4.2 + 6.5 = 10.7
115: 4.2 + 6.7 = 10.9
116: 4.2 + 9.6 = 13.8
117: 4.3 + 2.5 = 6.8
118: 4.3 + 2.8 = 7.1
119: 4.3 + 5.9 = 10.2
120: 4.5 + 1.7 = 6.2
121: 4.5 + 2.3 = 6.8
122: 4.5 + 2.6 = 7.1
123: 4.5 + 3.7 = 8.2
124: 4.5 + 6.2 = 10.7
125: 4.5 + 6.3 = 10.8
126: 4.6 + 2.5 = 7.1
127: 4.6 + 2.9 = 7.5
128: 4.6 + 3.2 = 7.8
129: 4.6 + 3.5 = 8.1
130: 4.6 + 3.9 = 8.5
131: 4.6 + 8.9 = 13.5
132: 4.7 + 1.2 = 5.9
133: 4.7 + 1.8 = 6.5
134: 4.7 + 3.5 = 8.2
135: 4.7 + 3.9 = 8.6
136: 4.7 + 5.6 = 10.3
137: 4.7 + 6.2 = 10.9
138: 4.7 + 8.5 = 13.2
139: 4.8 + 2.1 = 6.9
140: 4.8 + 2.3 = 7.1
141: 4.8 + 2.5 = 7.3
142: 4.8 + 7.5 = 12.3
143: 4.9 + 1.3 = 6.2
144: 4.9 + 5.3 = 10.2
145: 4.9 + 5.8 = 10.7
146: 4.9 + 8.6 = 13.5
147: 4.9 + 8.7 = 13.6
148: 5.1 + 3.6 = 8.7
149: 5.1 + 4.6 = 9.7
150: 5.1 + 4.7 = 9.8
151: 5.2 + 1.7 = 6.9
152: 5.2 + 3.4 = 8.6
153: 5.2 + 4.6 = 9.8
154: 5.2 + 7.8 = 13.0
155: 5.2 + 9.6 = 14.8
156: 5.3 + 6.7 = 12.0
157: 5.3 + 8.7 = 14.0
158: 5.3 + 8.9 = 14.2
159: 5.4 + 1.3 = 6.7
160: 5.4 + 1.8 = 7.2
161: 5.4 + 7.6 = 13.0
162: 5.4 + 8.2 = 13.6
163: 5.6 + 2.7 = 8.3
164: 5.6 + 3.1 = 8.7
165: 5.6 + 4.1 = 9.7
166: 5.6 + 4.2 = 9.8
167: 5.7 + 2.4 = 8.1
168: 5.7 + 2.6 = 8.3
169: 5.7 + 3.4 = 9.1
170: 5.7 + 4.6 = 10.3
171: 5.7 + 8.3 = 14.0
172: 5.8 + 1.6 = 7.4
173: 5.8 + 3.4 = 9.2
174: 5.8 + 3.6 = 9.4
175: 5.8 + 7.4 = 13.2
176: 5.9 + 1.3 = 7.2
177: 5.9 + 4.8 = 10.7
178: 5.9 + 6.4 = 12.3
179: 5.9 + 6.8 = 12.7
180: 6.1 + 2.4 = 8.5
181: 6.1 + 3.4 = 9.5
182: 6.2 + 1.3 = 7.5
183: 6.2 + 4.3 = 10.5
184: 6.2 + 8.3 = 14.5
185: 6.3 + 1.9 = 8.2
186: 6.3 + 4.2 = 10.5
187: 6.3 + 5.7 = 12.0
188: 6.3 + 8.2 = 14.5
189: 6.3 + 8.7 = 15.0
190: 6.3 + 9.4 = 15.7
191: 6.4 + 1.9 = 8.3
192: 6.4 + 2.1 = 8.5
193: 6.4 + 2.3 = 8.7
194: 6.4 + 3.1 = 9.5
195: 6.5 + 1.3 = 7.8
196: 6.5 + 1.4 = 7.9
197: 6.5 + 1.7 = 8.2
198: 6.5 + 3.2 = 9.7
199: 6.5 + 4.3 = 10.8
200: 6.5 + 7.8 = 14.3
201: 6.7 + 2.4 = 9.1
202: 6.7 + 2.8 = 9.5
203: 6.7 + 3.1 = 9.8
204: 6.7 + 8.3 = 15.0
205: 6.8 + 2.5 = 9.3
206: 6.9 + 3.5 = 10.4
207: 6.9 + 5.4 = 12.3
208: 6.9 + 5.8 = 12.7
209: 6.9 + 7.3 = 14.2
210: 7.2 + 1.4 = 8.6
211: 7.2 + 3.4 = 10.6
212: 7.2 + 5.8 = 13.0
213: 7.2 + 6.8 = 14.0
214: 7.3 + 1.2 = 8.5
215: 7.3 + 1.6 = 8.9
216: 7.3 + 2.5 = 9.8
217: 7.3 + 8.9 = 16.2
218: 7.3 + 9.2 = 16.5
219: 7.4 + 2.1 = 9.5
220: 7.4 + 3.5 = 10.9
221: 7.4 + 5.6 = 13.0
222: 7.4 + 6.5 = 13.9
223: 7.4 + 8.9 = 16.3
224: 7.5 + 1.8 = 9.3
225: 7.5 + 2.8 = 10.3
226: 7.5 + 2.9 = 10.4
227: 7.5 + 3.4 = 10.9
228: 7.5 + 6.4 = 13.9
229: 7.5 + 6.8 = 14.3
230: 7.5 + 9.3 = 16.8
231: 7.6 + 1.8 = 9.4
232: 7.6 + 2.9 = 10.5
233: 7.6 + 3.2 = 10.8
234: 7.6 + 4.9 = 12.5
235: 7.8 + 1.4 = 9.2
236: 7.8 + 2.6 = 10.4
237: 7.8 + 4.5 = 12.3
238: 7.8 + 5.4 = 13.2
239: 7.8 + 6.2 = 14.0
240: 7.9 + 2.4 = 10.3
241: 7.9 + 2.5 = 10.4
242: 7.9 + 2.6 = 10.5
243: 7.9 + 4.6 = 12.5
244: 7.9 + 6.3 = 14.2
245: 7.9 + 8.4 = 16.3
246: 8.2 + 6.5 = 14.7
247: 8.2 + 7.4 = 15.6
248: 8.2 + 9.4 = 17.6
249: 8.3 + 1.2 = 9.5
250: 8.3 + 2.6 = 10.9
251: 8.3 + 4.6 = 12.9
252: 8.3 + 7.6 = 15.9
253: 8.4 + 2.5 = 10.9
254: 8.4 + 6.9 = 15.3
255: 8.5 + 1.2 = 9.7
256: 8.5 + 3.9 = 12.4
257: 8.5 + 6.2 = 14.7
258: 8.5 + 7.9 = 16.4
259: 8.6 + 3.9 = 12.5
260: 8.6 + 5.7 = 14.3
261: 8.7 + 3.9 = 12.6
262: 8.7 + 4.5 = 13.2
263: 8.7 + 5.9 = 14.6
264: 8.9 + 3.6 = 12.5
265: 8.9 + 6.3 = 15.2
266: 8.9 + 6.4 = 15.3
267: 8.9 + 7.5 = 16.4
268: 9.2 + 4.5 = 13.7
269: 9.2 + 8.3 = 17.5
270: 9.2 + 8.4 = 17.6
271: 9.3 + 7.2 = 16.5
272: 9.3 + 8.2 = 17.5
273: 9.5 + 4.2 = 13.7
274: 9.5 + 7.3 = 16.8
Source: Shaun Errichiello
The way the problem is posted needs to show a decimal between the top two boxes. Otherwise it seems like we are adding a two digit whole number to the decimal.
Thanks for the correction. The problem has been updated.
273: 9,5 + 4,2 = 13,7
Right!
3.5+8.9=12.4
1.2+5.7=6.9
3.4+4.0=7.0
It equals 7.5, but alright.
3.8 + 5.2 =9.0
2.8+3.2=6.0
5.2+3.5=8.7
2.8+1.9=4.7
5.1=7.9=13
I found a different solution. 5.6 + 7.3 = 12.9
7.4*
it equals 7.4
i no under sand dis. plz help
The solutions given in the solutions section have a bit of a problem.
For example:
2.4 + 1.5 = 3.9 and 2.5 + 1.4 = 3.9 are both included, which you could argue are not unique because for every solution a.b + c.d = e.f, the equation a.d + c.b = e.f holds too (in other words you can swap the tenth places of the addends to get another solution)
This wouldn’t be a problem if you always included both. However, the list of solutions contains 2.9 + 1.8 = 4.7, but not 2.8 + 1.9 = 4.7. It contains both 3.2 + 1.7 = 4.9 and 3.7 + 1.2 = 4.9, but out of 1.2 + 3.6 = 4.8 and 1.6 + 3.2 = 4.8 it only contains the former.
So here’s the actual list of the 171 unique solutions (you can multiply that by 4 because you can swap the whole places of the addends or the tenth places of the addends to get another solution):
First the 71 solutions without leading zeros:
2.3 + 8.4 = 10.7
2.3 + 8.6 = 10.9
2.4 + 7.9 = 10.3
2.4 + 8.5 = 10.9
2.5 + 7.8 = 10.3
2.5 + 7.9 = 10.4
2.6 + 7.8 = 10.4
2.6 + 7.9 = 10.5
3.2 + 7.4 = 10.6
3.2 + 7.6 = 10.8
3.4 + 6.8 = 10.2
3.4 + 7.5 = 10.9
3.4 + 8.6 = 12.0
3.5 + 6.7 = 10.2
3.5 + 6.9 = 10.4
3.5 + 8.9 = 12.4
3.6 + 8.9 = 12.5
3.7 + 6.8 = 10.5
3.7 + 8.9 = 12.6
3.8 + 6.9 = 10.7
4.2 + 6.3 = 10.5
4.2 + 6.5 = 10.7
4.2 + 6.7 = 10.9
4.2 + 9.5 = 13.7
4.2 + 9.6 = 13.8
4.3 + 5.9 = 10.2
4.3 + 6.5 = 10.8
4.3 + 8.6 = 12.9
4.5 + 7.8 = 12.3
4.5 + 8.7 = 13.2
4.6 + 5.7 = 10.3
4.6 + 7.9 = 12.5
4.6 + 8.9 = 13.5
4.7 + 5.9 = 10.6
4.7 + 8.9 = 13.6
4.8 + 5.9 = 10.7
5.2 + 7.8 = 13.0
5.2 + 8.4 = 13.6
5.2 + 8.7 = 13.9
5.2 + 9.6 = 14.8
5.3 + 6.7 = 12.0
5.3 + 7.6 = 12.9
5.3 + 8.7 = 14.0
5.3 + 8.9 = 14.2
5.4 + 6.9 = 12.3
5.4 + 7.6 = 13.0
5.4 + 7.8 = 13.2
5.6 + 7.8 = 13.4
5.6 + 8.7 = 14.3
5.7 + 8.9 = 14.6
5.8 + 6.9 = 12.7
6.2 + 7.8 = 14.0
6.2 + 8.3 = 14.5
6.2 + 8.5 = 14.7
6.2 + 8.7 = 14.9
6.3 + 7.9 = 14.2
6.3 + 8.7 = 15.0
6.3 + 8.9 = 15.2
6.3 + 9.4 = 15.7
6.4 + 7.5 = 13.9
6.4 + 8.9 = 15.3
6.5 + 7.8 = 14.3
7.2 + 8.4 = 15.6
7.2 + 9.3 = 16.5
7.3 + 8.6 = 15.9
7.3 + 8.9 = 16.2
7.3 + 9.5 = 16.8
7.4 + 8.9 = 16.3
7.5 + 8.9 = 16.4
8.2 + 9.3 = 17.5
8.2 + 9.4 = 17.6
And then the 100 solutions with leading zeros:
1.2 + 3.5 = 04.7
1.2 + 3.6 = 04.8
1.2 + 3.7 = 04.9
1.2 + 4.6 = 05.8
1.2 + 4.7 = 05.9
1.2 + 5.7 = 06.9
1.2 + 6.3 = 07.5
1.2 + 7.3 = 08.5
1.2 + 7.4 = 08.6
1.2 + 8.3 = 09.5
1.2 + 8.4 = 09.6
1.2 + 8.5 = 09.7
1.3 + 4.6 = 05.9
1.3 + 4.9 = 06.2
1.3 + 5.4 = 06.7
1.3 + 5.9 = 07.2
1.3 + 6.5 = 07.8
1.3 + 6.9 = 08.2
1.3 + 7.6 = 08.9
1.3 + 8.4 = 09.7
1.4 + 2.5 = 03.9
1.4 + 3.8 = 05.2
1.4 + 5.8 = 07.2
1.4 + 5.9 = 07.3
1.4 + 6.5 = 07.9
1.4 + 6.9 = 08.3
1.4 + 7.5 = 08.9
1.4 + 7.8 = 09.2
1.5 + 2.8 = 04.3
1.5 + 4.7 = 06.2
1.5 + 4.8 = 06.3
1.5 + 6.7 = 08.2
1.5 + 6.9 = 08.4
1.5 + 7.8 = 09.3
1.6 + 2.7 = 04.3
1.6 + 2.9 = 04.5
1.6 + 3.8 = 05.4
1.6 + 5.8 = 07.4
1.6 + 7.8 = 09.4
1.7 + 2.8 = 04.5
1.7 + 2.9 = 04.6
1.7 + 3.9 = 05.6
1.7 + 4.8 = 06.5
1.8 + 2.9 = 04.7
1.8 + 3.9 = 05.7
1.8 + 4.9 = 06.7
2.1 + 3.6 = 05.7
2.1 + 3.7 = 05.8
2.1 + 3.8 = 05.9
2.1 + 4.7 = 06.8
2.1 + 4.8 = 06.9
2.1 + 5.3 = 07.4
2.1 + 5.8 = 07.9
2.1 + 6.3 = 08.4
2.1 + 6.4 = 08.5
2.1 + 7.3 = 09.4
2.1 + 7.4 = 09.5
2.1 + 7.5 = 09.6
2.3 + 4.5 = 06.8
2.3 + 4.8 = 07.1
2.3 + 5.6 = 07.9
2.3 + 6.4 = 08.7
2.3 + 6.8 = 09.1
2.3 + 7.5 = 09.8
2.4 + 3.7 = 06.1
2.4 + 5.7 = 08.1
2.4 + 5.9 = 08.3
2.4 + 6.5 = 08.9
2.4 + 6.7 = 09.1
2.5 + 3.9 = 06.4
2.5 + 4.6 = 07.1
2.5 + 4.8 = 07.3
2.5 + 6.8 = 09.3
2.6 + 4.9 = 07.5
2.6 + 5.7 = 08.3
2.7 + 3.8 = 06.5
2.7 + 5.9 = 08.6
2.7 + 6.8 = 09.5
2.8 + 3.9 = 06.7
3.1 + 4.5 = 07.6
3.1 + 4.8 = 07.9
3.1 + 5.6 = 08.7
3.1 + 6.4 = 09.5
3.1 + 6.7 = 09.8
3.2 + 4.6 = 07.8
3.2 + 4.9 = 08.1
3.2 + 5.4 = 08.6
3.2 + 5.7 = 08.9
3.2 + 6.5 = 09.7
3.4 + 5.7 = 09.1
3.4 + 5.8 = 09.2
3.5 + 4.6 = 08.1
3.5 + 4.7 = 08.2
3.6 + 4.9 = 08.5
3.6 + 5.8 = 09.4
3.7 + 4.9 = 08.6
4.1 + 5.2 = 09.3
4.1 + 5.6 = 09.7
4.1 + 5.7 = 09.8
4.2 + 5.6 = 09.8
what is 10.3+10.8 adding using tenths