Open Middle®
Directions: Using the digits 0 through 9 at most one time, fill in the boxes to make the sum of the interior angles of a triangle.
Hint
Answer
Number of Unique Solutions: 192
1: 12 + 73 + 95 = 180
2: 13 + 72 + 95 = 180
3: 13 + 92 + 75 = 180
4: 15 + 67 + 98 = 180
5: 15 + 76 + 89 = 180
6: 15 + 79 + 86 = 180
7: 15 + 97 + 68 = 180
8: 17 + 65 + 98 = 180
9: 18 + 95 + 67 = 180
10: 18 + 97 + 65 = 180
11: 19 + 85 + 76 = 180
12: 19 + 86 + 75 = 180
13: 21 + 64 + 95 = 180
14: 21 + 74 + 85 = 180
15: 21 + 75 + 84 = 180
16: 21 + 76 + 83 = 180
17: 21 + 86 + 73 = 180
18: 21 + 94 + 65 = 180
19: 23 + 71 + 86 = 180
20: 24 + 67 + 89 = 180
21: 24 + 75 + 81 = 180
22: 24 + 85 + 71 = 180
23: 24 + 87 + 69 = 180
24: 24 + 91 + 65 = 180
25: 24 + 95 + 61 = 180
26: 25 + 61 + 94 = 180
27: 25 + 64 + 91 = 180
28: 25 + 81 + 74 = 180
29: 25 + 84 + 71 = 180
30: 26 + 71 + 83 = 180
31: 26 + 81 + 73 = 180
32: 27 + 89 + 64 = 180
33: 31 + 67 + 82 = 180
34: 31 + 85 + 64 = 180
35: 31 + 87 + 62 = 180
36: 31 + 92 + 57 = 180
37: 32 + 57 + 91 = 180
38: 32 + 81 + 67 = 180
39: 32 + 87 + 61 = 180
40: 32 + 97 + 51 = 180
41: 34 + 57 + 89 = 180
42: 34 + 61 + 85 = 180
43: 34 + 87 + 59 = 180
44: 35 + 64 + 81 = 180
45: 35 + 98 + 47 = 180
46: 37 + 48 + 95 = 180
47: 37 + 61 + 82 = 180
48: 37 + 89 + 54 = 180
49: 37 + 91 + 52 = 180
50: 37 + 92 + 51 = 180
51: 38 + 47 + 95 = 180
52: 38 + 97 + 45 = 180
53: 39 + 54 + 87 = 180
54: 39 + 57 + 84 = 180
55: 41 + 53 + 86 = 180
56: 41 + 57 + 82 = 180
57: 41 + 87 + 52 = 180
58: 42 + 65 + 73 = 180
59: 43 + 56 + 81 = 180
60: 43 + 58 + 79 = 180
61: 43 + 59 + 78 = 180
62: 43 + 75 + 62 = 180
63: 43 + 86 + 51 = 180
64: 45 + 62 + 73 = 180
65: 45 + 98 + 37 = 180
66: 46 + 53 + 81 = 180
67: 46 + 83 + 51 = 180
68: 47 + 38 + 95 = 180
69: 48 + 59 + 73 = 180
70: 48 + 97 + 35 = 180
71: 49 + 53 + 78 = 180
72: 49 + 58 + 73 = 180
73: 51 + 37 + 92 = 180
74: 51 + 82 + 47 = 180
75: 51 + 87 + 42 = 180
76: 52 + 81 + 47 = 180
77: 52 + 97 + 31 = 180
78: 53 + 41 + 86 = 180
79: 53 + 49 + 78 = 180
80: 53 + 79 + 48 = 180
81: 54 + 37 + 89 = 180
82: 54 + 39 + 87 = 180
83: 56 + 81 + 43 = 180
84: 56 + 83 + 41 = 180
85: 57 + 32 + 91 = 180
86: 57 + 39 + 84 = 180
87: 57 + 81 + 42 = 180
88: 57 + 82 + 41 = 180
89: 57 + 89 + 34 = 180
90: 58 + 49 + 73 = 180
91: 58 + 79 + 43 = 180
92: 59 + 48 + 73 = 180
93: 59 + 78 + 43 = 180
94: 59 + 84 + 37 = 180
95: 61 + 25 + 94 = 180
96: 61 + 32 + 87 = 180
97: 61 + 34 + 85 = 180
98: 61 + 35 + 84 = 180
99: 61 + 82 + 37 = 180
100: 62 + 45 + 73 = 180
101: 62 + 81 + 37 = 180
102: 63 + 45 + 72 = 180
103: 63 + 75 + 42 = 180
104: 64 + 29 + 87 = 180
105: 64 + 35 + 81 = 180
106: 65 + 17 + 98 = 180
107: 65 + 18 + 97 = 180
108: 65 + 42 + 73 = 180
109: 65 + 72 + 43 = 180
110: 65 + 81 + 34 = 180
111: 65 + 84 + 31 = 180
112: 67 + 18 + 95 = 180
113: 67 + 24 + 89 = 180
114: 67 + 31 + 82 = 180
115: 67 + 84 + 29 = 180
116: 68 + 95 + 17 = 180
117: 69 + 27 + 84 = 180
118: 71 + 24 + 85 = 180
119: 71 + 26 + 83 = 180
120: 71 + 86 + 23 = 180
121: 72 + 13 + 95 = 180
122: 72 + 63 + 45 = 180
123: 72 + 65 + 43 = 180
124: 72 + 93 + 15 = 180
125: 73 + 12 + 95 = 180
126: 73 + 92 + 15 = 180
127: 74 + 21 + 85 = 180
128: 75 + 16 + 89 = 180
129: 75 + 21 + 84 = 180
130: 75 + 43 + 62 = 180
131: 75 + 63 + 42 = 180
132: 75 + 93 + 12 = 180
133: 76 + 19 + 85 = 180
134: 76 + 81 + 23 = 180
135: 76 + 89 + 15 = 180
136: 79 + 53 + 48 = 180
137: 79 + 85 + 16 = 180
138: 79 + 86 + 15 = 180
139: 81 + 23 + 76 = 180
140: 81 + 24 + 75 = 180
141: 81 + 32 + 67 = 180
142: 81 + 37 + 62 = 180
143: 81 + 46 + 53 = 180
144: 81 + 52 + 47 = 180
145: 81 + 57 + 42 = 180
146: 81 + 65 + 34 = 180
147: 81 + 73 + 26 = 180
148: 81 + 74 + 25 = 180
149: 82 + 51 + 47 = 180
150: 83 + 21 + 76 = 180
151: 83 + 46 + 51 = 180
152: 83 + 56 + 41 = 180
153: 84 + 25 + 71 = 180
154: 84 + 59 + 37 = 180
155: 84 + 61 + 35 = 180
156: 84 + 65 + 31 = 180
157: 84 + 67 + 29 = 180
158: 84 + 69 + 27 = 180
159: 85 + 64 + 31 = 180
160: 85 + 79 + 16 = 180
161: 86 + 19 + 75 = 180
162: 86 + 43 + 51 = 180
163: 86 + 73 + 21 = 180
164: 87 + 41 + 52 = 180
165: 87 + 42 + 51 = 180
166: 87 + 59 + 34 = 180
167: 87 + 62 + 31 = 180
168: 87 + 64 + 29 = 180
169: 87 + 69 + 24 = 180
170: 89 + 27 + 64 = 180
171: 89 + 75 + 16 = 180
172: 91 + 25 + 64 = 180
173: 91 + 37 + 52 = 180
174: 91 + 65 + 24 = 180
175: 92 + 13 + 75 = 180
176: 92 + 31 + 57 = 180
177: 92 + 73 + 15 = 180
178: 93 + 72 + 15 = 180
179: 93 + 75 + 12 = 180
180: 94 + 21 + 65 = 180
181: 95 + 21 + 64 = 180
182: 95 + 37 + 48 = 180
183: 95 + 61 + 24 = 180
184: 95 + 68 + 17 = 180
185: 97 + 15 + 68 = 180
186: 97 + 32 + 51 = 180
187: 97 + 45 + 38 = 180
188: 97 + 48 + 35 = 180
189: 97 + 52 + 31 = 180
190: 98 + 15 + 67 = 180
191: 98 + 37 + 45 = 180
192: 98 + 47 + 35 = 180
Source: Ashley Henderson
Once given the hint, the problem doesn’t really do much for furthering students’ understanding of the triangle sum theorem, but my students had a great time making sense of the problem and then persevering to find as many answers as possible. Used in an intervention class and overheard a student say, “It’s been a while since I’ve had this much fun in math class” (since I made it a competition) and another student agree.
67+89+24
67+89+24
89+24+67
67+89+24
67+89+24
42+65+71
67+89+24
60+60+60
80+79+21
67+89+24
89+24+67
56+34+90
61+56+63
99+50+31
60 + 60 + 60
60+60+60 because u haft to divide 180/3
67 + 89 + 24
67+89+24
80+74+26
10,80,90
59+60+61
67+89+24
67+89+24
67+89+24
80+50+50
49+60+71
38+52+90
67 90 23
69+87+24
67+89+24
67+89+24
60+60+60
90+45+45
35+68+77
67+89++24
67+89+24=180
91 + 25 + 64
48 + 72 + 60 =180
69+69+42
94+66+20
59+37+84
45 60 75
34+67+90
60+60+60
60 60 60
13+72+95=180
30+59+91
Maybe as an extension you could ask students to use a protractor to draw a triangle with the same 3 angles they came up with? I agree, it’s more of an addition puzzle than a triangle sum puzzle as is, but it’s a real fun one for the start of the year and to introduce them to open middle! 🙂
67+89+24
67 + 89 + 24
I got it correct
40+50+90=180 degrees
67+89+24
80+74+26
60+70+50
67+89+24 that what I think
67+89+24
67+70+43
72+38+70
80 + 26 + 74 = 180°
74 + 26 + 80 =180 degrees
60+60+60=180
63 + 53 + 64=180
80+74+26=180
80+74+26=180
67+89+24=180
67+89+24
67+89+24 is the answer.
I have an another one. 30+75+75
60 plus 60 plus 60
21+54+13=180
90+24+66…pretty sure right …
80+57+43=180
80+57+43
First, I used a random number to use as my base. Then that helped me narrow it down to help me find the other two numbers that would add up to 180. All I had to do was subtract my number from 180 and find 2 numbers that would have a sum equal to my number subtracted by 180.
67+89+24 =180
67+89+24
80+74+26
I got 34+85+61=180
67+89+24
80+74+26
67+89+24
90+40+50
90+35+45
I found the answer 80 + 24 + 76.
67+89+24=180
60+60+60
56+25+99
60+120=180
23+51+74
60+60+60
30+30+120=180
60+80+40=180
80 + 74 + 26
67+89+24
01+97+82
60 + 60 + 60 = 180
75+25+60=180
60+60+60
60+60+60=180
i got…58+32+90=180
i got 76+10+94=180
nevermind! i got 40+60+80
60+70+50=180
I got 81+75+24
67+89+24
67+89+24 is what I got.
60+60+60
67+89+24
90+60+30=180 degrees
86+70+24
45+45+45 is what i got
67+89+24 is what i got
52+90+38
67+89+24
80+21+79
67+89+24
Suleman
90+52+38
89+23+68
90+36+54
40+70+70
47+52+81=180
67+35+78=180
90+40+50=180
60+60+60
60+60+60=180
178+1+1
67 + 89 +24 =180
67 + 89 + 24=180
67+89+24=180 is the answer i got.
30 +60 +90 is the answer.
67+89=156+24=180
100+40+40
34+65+81
61+62+57
80+74+26=180
41+20+37
85 + 31 + 64 = 180 and no numbers are repeated.
67 + 89 + 24
67+89+24
67 + 89+ 24
45+45+90
67+89+24
80+74+26=180
24+76+26
24+67+89=180
90+60+30=180
60°+60°+60°=180°
My answer is 37 + 67 + 76
I got this by thinking of a random number which was 37 and I subtracted that by 180 and I got 153, then I subtract 67 with 153 and got 76, Finally I subtract 76 with 76 and got 0. To check my answer I added them all together and got 180.
40°+40°+100°=180
45 + 45 + 90 = 180
45+45+90=180
52+50+78
104+30+46=180
74+69+37=180
67+89+24
89+67+24
60+60+60
60+89+24
98+64+17
76+84+20
32+58+90=180
80+49+51=180
80+39+61
67+89+24
81+64+35=180
I got 34+84+19+
34+84+19+
50+69+61=180
67+89+24
70+10+100=180
60+89+24
50+61+69= 180 degress
91+75+14=180
91+75+14
67+89+24
67+22+78
67+89+24
67+89+24
67+89+24
39+57+84
67+89+24.
92+50+38=180
60+60+60=180
80+74+26
67+89+29
80+74+26=180
80+80+20=180
67+89+24
45 + 45 + 90
45+45+90
80+80+20=180
27 + 3 + 150 = 180
67 + 89 + 24
67 + 89 + 24=180
60 +60+60=180
67+89+24=180
100 +40+40=180
67+89+24=180
67+89+27=180
67 91 22
60+60+60
89+67+24
67+89+24=180
60 + 48 + 72 = 180
57+96+27=180
80+80+20 180
53 + 82+ 45
24+67+89
60+71+49=180
66+41+73
89+24+67
60+70+50
40+50+90=180 degrees
50+50+80=180 degrees
60+60+60=180
70+50+60=180
53 + 82 + 45
60+60+60
The answer is 24+67+89
23+90+67
60 + 60 + 60 = 180
60+60+60=180
60+60+60=180
64+85+31
28+98+54
50+45+85
67+89+24
99+62+19
15+79+86
Three numbers that add up to 180 degrees are 91,24, 64 degrees. I got to this answer by starting with the largest number closes to 100 so therefore I add a different combination of numbers that would get me to one 180 degrees.
52 + 90 + 38 = 180
60 degrees + 60 degrees + 60 degrees = 180 degrees
85+45+50=180
67+89+24
12+95+73=180
85 + 64 + 31 = 180
45+45+90=180
12+73+95=180
62+45+72 = 180
70+50+60
17+65+98=180
67+89+24=180
90+36+54
60+50+70
Im suprised there are about 200 answers.
34+56+90=180
178+1+1
130+30+20