Open Middle®
Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the sum of the interior angles of a triangle.
Hint
What is the sum of the interior angle of a triangle?
Answer
There are 192 unique solutions. Three examples are:
1: 12 + 73 + 95 = 180
2: 13 + 72 + 95 = 180
3: 13 + 92 + 75 = 180
1: 12 + 73 + 95 = 180
2: 13 + 72 + 95 = 180
3: 13 + 92 + 75 = 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