Need help on math.

godhatesjustyou

godhatesjustyou

CAGiversary!
Feedback
26 (100%)
my math teacher gave us 2 problems to do over spring break (which starts on friday) and since i'm not all too much of a math whiz, i'd like to ask you guys. (after seeing that '0.999~=1' deal, you guys must be good)

1)How many non-similar interger-angled triangles are there?


2)How many numbers less than one million have at least one zero as a digit?


i'm assuming that on number 2, my teacher meant to say "non-negative numbers" somewhere in there, otherwise it'd be infinite.

also, my friend and i worked on number 2 for the whole period, and this was the answer we came up with, just wanted to see if we got close to the real answer: 402,530.

thanks to all who are willing to help.

-edit- oh yeah, please post how you got the answer also. :D
 
woah man, i'm all for helping a cheapass gamer on his math homework in the whee hours of the morning, but my brain doesn't fully function this early and i can't dedicate hours of my time figuring out how many zero's there are in numbers 1- a million.
and i didn't even look at the ~999. = 1 thread. (it's wrong right? :p exactly)
 
Yeah, I pasted the completely wrong thing earlier, but actually fixed it before your comment. So we're agreed we have the right numbers now?
 
ok, using 3 digits as an example since it's relatively easier yet descriptive:

we have xyz, let's say z = 0 first
x = 9 possibilities
y = 10 possibilities
so we have 90 total if z is 0

now let's say y = 0
x = 9 pos.
z = (10 - 1) pos. (-1 because we don't want 100, 200, etc, twice)
so that's 81 total if y is 0

so for 3 digits I believe it's 171

it looks more complicated when you get to bigger numbers since there are more dupes. Can't think of any easy formulas for this at the moment.
 
Oh, I'm sorry, AT LEAST one zero?

OK, let's work it out, give me a couple of minutes. I should definitely not post before 6AM.
 
Let's flip the question around:

How many numbers less than one million have absolutely NO zeroes as a digit?

1 digit numbers: 9
2 digit numbers: 9^2
etc etc
So the result for (0-999,999) is 1,000,000 -9 -9^2 -9^3 -9^4 -9^5 -9^6 = 402130.

Add one if you count 1 million in the range.
 
eldad9 got the easiest way to do #2 by looking at it backwards. It makes the assumption that you are including 0 which will depend on whether the teacher was assuming positive or non-negative as you mentioned. #1 is a bit more involved. This should get you started though.

The sum of the angles of any triangle is 180 degrees.

Start with the first angle and call it x. Valid integer values for x range from 1 to 178. (Yes, some of these triangles will look very bizarre.)

Call the second angle y. For each value of x, valid values for y range from 1 to 179-x.

Call the third angle z. It's value is fixed to a single value by the values of x and y (z=180-x-y) so it doesn't add to the number of triangles.

Assuming my ASCII art comes out you get the following summation:

178 179-x
------ ------
\ \
/ / 1
------ ------
x=1 y=1

Which gives you the total number of triangles. You can use the trick of the sum from 1 to n = n*(n+1)/2 to turn it into

178
------
\
/ (179-x)*(180-x)/2
------
x=1


but you're still not going to work it out by hand.

Then you've got the requirement for non-similar triangles which is almost but not quite as easy as dividing by 3. :wink:

Enjoy working out the details. Hope you have a handy copy of Mathematica or something similar through your school. This one is a real bear.
 
oh wow, thanks guys. i most definitely know how to do number 2 (think outta the box, duh), but number 1 is gonna take some time. once again, thank you eldad9 and roland_htg. and sorry for letting you guys work your brain so early in the morning, i posted this before going to sleep.
 
If X = 178 how many solutions do you have for y and z? Answer = 1 (y=1, z=1)

X=177? Answer = 1 (y=1, z=2) Because y=2, z=1 is similar

X= 176? Answer = 2 (1,3) (2,2)
x = 175? Answer = 2 (1,4) (2,3)
174? 3.
173? 3.
And so on until you can have another angle in the triangle that = x (which first happens at X = 89 (x=89, y=89, z=2)) making all other values of x similar to triangles you've already counted. Which means you have:
((179-89)/2) 45 pairs of like results (i.e. X= 178 and X=177 are like results of 1). So your answer is:
2 * the sum(1->45)
or 2* ((45*46)/2)
or 45*46
or 2070 non-similar integer-angled triangles



But I could be wrong....


Editorial Opinion: You really should be trying to do these on your own. Math questions like this are great ways of helping you think more analytically, rather than slogging through repetitive calculations. I know that you could write a small program in a computer to brute force this problem in just a few minutes, but what if the problem were more complex? Yes, I can add 1+2+3+4+5+6+7+8+9+10 and get 55, but what if you wanted me to sum the number 1 to 487? Break the problem down, find a general solution, and apply it to the larger problem. There are problems large enough that brute forcing will take even a computer hundreds of years to finish, but can be solved easily by hand if you can come up with an equation to do it.

It's the process that's important, not the answer.

Okay, that's enough ranting from me.
 
Yeah, that backwards solution is a great way to solve #2 and many other math problems similar to that. If you go to math competitions regularly or are on the math team, you see these kinds of questions all the time.
 
I've got 2700:

1 1 178
1 2 177
1 3 176
1 4 175
1 5 174
1 6 173
1 7 172
1 8 171
1 9 170
1 10 169
1 11 168
1 12 167
1 13 166
1 14 165
1 15 164
1 16 163
1 17 162
1 18 161
1 19 160
1 20 159
1 21 158
1 22 157
1 23 156
1 24 155
1 25 154
1 26 153
1 27 152
1 28 151
1 29 150
1 30 149
1 31 148
1 32 147
1 33 146
1 34 145
1 35 144
1 36 143
1 37 142
1 38 141
1 39 140
1 40 139
1 41 138
1 42 137
1 43 136
1 44 135
1 45 134
1 46 133
1 47 132
1 48 131
1 49 130
1 50 129
1 51 128
1 52 127
1 53 126
1 54 125
1 55 124
1 56 123
1 57 122
1 58 121
1 59 120
1 60 119
1 61 118
1 62 117
1 63 116
1 64 115
1 65 114
1 66 113
1 67 112
1 68 111
1 69 110
1 70 109
1 71 108
1 72 107
1 73 106
1 74 105
1 75 104
1 76 103
1 77 102
1 78 101
1 79 100
1 80 99
1 81 98
1 82 97
1 83 96
1 84 95
1 85 94
1 86 93
1 87 92
1 88 91
1 89 90
2 2 176
2 3 175
2 4 174
2 5 173
2 6 172
2 7 171
2 8 170
2 9 169
2 10 168
2 11 167
2 12 166
2 13 165
2 14 164
2 15 163
2 16 162
2 17 161
2 18 160
2 19 159
2 20 158
2 21 157
2 22 156
2 23 155
2 24 154
2 25 153
2 26 152
2 27 151
2 28 150
2 29 149
2 30 148
2 31 147
2 32 146
2 33 145
2 34 144
2 35 143
2 36 142
2 37 141
2 38 140
2 39 139
2 40 138
2 41 137
2 42 136
2 43 135
2 44 134
2 45 133
2 46 132
2 47 131
2 48 130
2 49 129
2 50 128
2 51 127
2 52 126
2 53 125
2 54 124
2 55 123
2 56 122
2 57 121
2 58 120
2 59 119
2 60 118
2 61 117
2 62 116
2 63 115
2 64 114
2 65 113
2 66 112
2 67 111
2 68 110
2 69 109
2 70 108
2 71 107
2 72 106
2 73 105
2 74 104
2 75 103
2 76 102
2 77 101
2 78 100
2 79 99
2 80 98
2 81 97
2 82 96
2 83 95
2 84 94
2 85 93
2 86 92
2 87 91
2 88 90
2 89 89
3 3 174
3 4 173
3 5 172
3 6 171
3 7 170
3 8 169
3 9 168
3 10 167
3 11 166
3 12 165
3 13 164
3 14 163
3 15 162
3 16 161
3 17 160
3 18 159
3 19 158
3 20 157
3 21 156
3 22 155
3 23 154
3 24 153
3 25 152
3 26 151
3 27 150
3 28 149
3 29 148
3 30 147
3 31 146
3 32 145
3 33 144
3 34 143
3 35 142
3 36 141
3 37 140
3 38 139
3 39 138
3 40 137
3 41 136
3 42 135
3 43 134
3 44 133
3 45 132
3 46 131
3 47 130
3 48 129
3 49 128
3 50 127
3 51 126
3 52 125
3 53 124
3 54 123
3 55 122
3 56 121
3 57 120
3 58 119
3 59 118
3 60 117
3 61 116
3 62 115
3 63 114
3 64 113
3 65 112
3 66 111
3 67 110
3 68 109
3 69 108
3 70 107
3 71 106
3 72 105
3 73 104
3 74 103
3 75 102
3 76 101
3 77 100
3 78 99
3 79 98
3 80 97
3 81 96
3 82 95
3 83 94
3 84 93
3 85 92
3 86 91
3 87 90
3 88 89
4 4 172
4 5 171
4 6 170
4 7 169
4 8 168
4 9 167
4 10 166
4 11 165
4 12 164
4 13 163
4 14 162
4 15 161
4 16 160
4 17 159
4 18 158
4 19 157
4 20 156
4 21 155
4 22 154
4 23 153
4 24 152
4 25 151
4 26 150
4 27 149
4 28 148
4 29 147
4 30 146
4 31 145
4 32 144
4 33 143
4 34 142
4 35 141
4 36 140
4 37 139
4 38 138
4 39 137
4 40 136
4 41 135
4 42 134
4 43 133
4 44 132
4 45 131
4 46 130
4 47 129
4 48 128
4 49 127
4 50 126
4 51 125
4 52 124
4 53 123
4 54 122
4 55 121
4 56 120
4 57 119
4 58 118
4 59 117
4 60 116
4 61 115
4 62 114
4 63 113
4 64 112
4 65 111
4 66 110
4 67 109
4 68 108
4 69 107
4 70 106
4 71 105
4 72 104
4 73 103
4 74 102
4 75 101
4 76 100
4 77 99
4 78 98
4 79 97
4 80 96
4 81 95
4 82 94
4 83 93
4 84 92
4 85 91
4 86 90
4 87 89
4 88 88
5 5 170
5 6 169
5 7 168
5 8 167
5 9 166
5 10 165
5 11 164
5 12 163
5 13 162
5 14 161
5 15 160
5 16 159
5 17 158
5 18 157
5 19 156
5 20 155
5 21 154
5 22 153
5 23 152
5 24 151
5 25 150
5 26 149
5 27 148
5 28 147
5 29 146
5 30 145
5 31 144
5 32 143
5 33 142
5 34 141
5 35 140
5 36 139
5 37 138
5 38 137
5 39 136
5 40 135
5 41 134
5 42 133
5 43 132
5 44 131
5 45 130
5 46 129
5 47 128
5 48 127
5 49 126
5 50 125
5 51 124
5 52 123
5 53 122
5 54 121
5 55 120
5 56 119
5 57 118
5 58 117
5 59 116
5 60 115
5 61 114
5 62 113
5 63 112
5 64 111
5 65 110
5 66 109
5 67 108
5 68 107
5 69 106
5 70 105
5 71 104
5 72 103
5 73 102
5 74 101
5 75 100
5 76 99
5 77 98
5 78 97
5 79 96
5 80 95
5 81 94
5 82 93
5 83 92
5 84 91
5 85 90
5 86 89
5 87 88
6 6 168
6 7 167
6 8 166
6 9 165
6 10 164
6 11 163
6 12 162
6 13 161
6 14 160
6 15 159
6 16 158
6 17 157
6 18 156
6 19 155
6 20 154
6 21 153
6 22 152
6 23 151
6 24 150
6 25 149
6 26 148
6 27 147
6 28 146
6 29 145
6 30 144
6 31 143
6 32 142
6 33 141
6 34 140
6 35 139
6 36 138
6 37 137
6 38 136
6 39 135
6 40 134
6 41 133
6 42 132
6 43 131
6 44 130
6 45 129
6 46 128
6 47 127
6 48 126
6 49 125
6 50 124
6 51 123
6 52 122
6 53 121
6 54 120
6 55 119
6 56 118
6 57 117
6 58 116
6 59 115
6 60 114
6 61 113
6 62 112
6 63 111
6 64 110
6 65 109
6 66 108
6 67 107
6 68 106
6 69 105
6 70 104
6 71 103
6 72 102
6 73 101
6 74 100
6 75 99
6 76 98
6 77 97
6 78 96
6 79 95
6 80 94
6 81 93
6 82 92
6 83 91
6 84 90
6 85 89
6 86 88
6 87 87
7 7 166
7 8 165
7 9 164
7 10 163
7 11 162
7 12 161
7 13 160
7 14 159
7 15 158
7 16 157
7 17 156
7 18 155
7 19 154
7 20 153
7 21 152
7 22 151
7 23 150
7 24 149
7 25 148
7 26 147
7 27 146
7 28 145
7 29 144
7 30 143
7 31 142
7 32 141
7 33 140
7 34 139
7 35 138
7 36 137
7 37 136
7 38 135
7 39 134
7 40 133
7 41 132
7 42 131
7 43 130
7 44 129
7 45 128
7 46 127
7 47 126
7 48 125
7 49 124
7 50 123
7 51 122
7 52 121
7 53 120
7 54 119
7 55 118
7 56 117
7 57 116
7 58 115
7 59 114
7 60 113
7 61 112
7 62 111
7 63 110
7 64 109
7 65 108
7 66 107
7 67 106
7 68 105
7 69 104
7 70 103
7 71 102
7 72 101
7 73 100
7 74 99
7 75 98
7 76 97
7 77 96
7 78 95
7 79 94
7 80 93
7 81 92
7 82 91
7 83 90
7 84 89
7 85 88
7 86 87
8 8 164
8 9 163
8 10 162
8 11 161
8 12 160
8 13 159
8 14 158
8 15 157
8 16 156
8 17 155
8 18 154
8 19 153
8 20 152
8 21 151
8 22 150
8 23 149
8 24 148
8 25 147
8 26 146
8 27 145
8 28 144
8 29 143
8 30 142
8 31 141
8 32 140
8 33 139
8 34 138
8 35 137
8 36 136
8 37 135
8 38 134
8 39 133
8 40 132
8 41 131
8 42 130
8 43 129
8 44 128
8 45 127
8 46 126
8 47 125
8 48 124
8 49 123
8 50 122
8 51 121
8 52 120
8 53 119
8 54 118
8 55 117
8 56 116
8 57 115
8 58 114
8 59 113
8 60 112
8 61 111
8 62 110
8 63 109
8 64 108
8 65 107
8 66 106
8 67 105
8 68 104
8 69 103
8 70 102
8 71 101
8 72 100
8 73 99
8 74 98
8 75 97
8 76 96
8 77 95
8 78 94
8 79 93
8 80 92
8 81 91
8 82 90
8 83 89
8 84 88
8 85 87
8 86 86
9 9 162
9 10 161
9 11 160
9 12 159
9 13 158
9 14 157
9 15 156
9 16 155
9 17 154
9 18 153
9 19 152
9 20 151
9 21 150
9 22 149
9 23 148
9 24 147
9 25 146
9 26 145
9 27 144
9 28 143
9 29 142
9 30 141
9 31 140
9 32 139
9 33 138
9 34 137
9 35 136
9 36 135
9 37 134
9 38 133
9 39 132
9 40 131
9 41 130
9 42 129
9 43 128
9 44 127
9 45 126
9 46 125
9 47 124
9 48 123
9 49 122
9 50 121
9 51 120
9 52 119
9 53 118
9 54 117
9 55 116
9 56 115
9 57 114
9 58 113
9 59 112
9 60 111
9 61 110
9 62 109
9 63 108
9 64 107
9 65 106
9 66 105
9 67 104
9 68 103
9 69 102
9 70 101
9 71 100
9 72 99
9 73 98
9 74 97
9 75 96
9 76 95
9 77 94
9 78 93
9 79 92
9 80 91
9 81 90
9 82 89
9 83 88
9 84 87
9 85 86
10 10 160
10 11 159
10 12 158
10 13 157
10 14 156
10 15 155
10 16 154
10 17 153
10 18 152
10 19 151
10 20 150
10 21 149
10 22 148
10 23 147
10 24 146
10 25 145
10 26 144
10 27 143
10 28 142
10 29 141
10 30 140
10 31 139
10 32 138
10 33 137
10 34 136
10 35 135
10 36 134
10 37 133
10 38 132
10 39 131
10 40 130
10 41 129
10 42 128
10 43 127
10 44 126
10 45 125
10 46 124
10 47 123
10 48 122
10 49 121
10 50 120
10 51 119
10 52 118
10 53 117
10 54 116
10 55 115
10 56 114
10 57 113
10 58 112
10 59 111
10 60 110
10 61 109
10 62 108
10 63 107
10 64 106
10 65 105
10 66 104
10 67 103
10 68 102
10 69 101
10 70 100
10 71 99
10 72 98
10 73 97
10 74 96
10 75 95
10 76 94
10 77 93
10 78 92
10 79 91
10 80 90
10 81 89
10 82 88
10 83 87
10 84 86
10 85 85
11 11 158
11 12 157
11 13 156
11 14 155
11 15 154
11 16 153
11 17 152
11 18 151
11 19 150
11 20 149
11 21 148
11 22 147
11 23 146
11 24 145
11 25 144
11 26 143
11 27 142
11 28 141
11 29 140
11 30 139
11 31 138
11 32 137
11 33 136
11 34 135
11 35 134
11 36 133
11 37 132
11 38 131
11 39 130
11 40 129
11 41 128
11 42 127
11 43 126
11 44 125
11 45 124
11 46 123
11 47 122
11 48 121
11 49 120
11 50 119
11 51 118
11 52 117
11 53 116
11 54 115
11 55 114
11 56 113
11 57 112
11 58 111
11 59 110
11 60 109
11 61 108
11 62 107
11 63 106
11 64 105
11 65 104
11 66 103
11 67 102
11 68 101
11 69 100
11 70 99
11 71 98
11 72 97
11 73 96
11 74 95
11 75 94
11 76 93
11 77 92
11 78 91
11 79 90
11 80 89
11 81 88
11 82 87
11 83 86
11 84 85
12 12 156
12 13 155
12 14 154
12 15 153
12 16 152
12 17 151
12 18 150
12 19 149
12 20 148
12 21 147
12 22 146
12 23 145
12 24 144
12 25 143
12 26 142
12 27 141
12 28 140
12 29 139
12 30 138
12 31 137
12 32 136
12 33 135
12 34 134
12 35 133
12 36 132
12 37 131
12 38 130
12 39 129
12 40 128
12 41 127
12 42 126
12 43 125
12 44 124
12 45 123
12 46 122
12 47 121
12 48 120
12 49 119
12 50 118
12 51 117
12 52 116
12 53 115
12 54 114
12 55 113
12 56 112
12 57 111
12 58 110
12 59 109
12 60 108
12 61 107
12 62 106
12 63 105
12 64 104
12 65 103
12 66 102
12 67 101
12 68 100
12 69 99
12 70 98
12 71 97
12 72 96
12 73 95
12 74 94
12 75 93
12 76 92
12 77 91
12 78 90
12 79 89
12 80 88
12 81 87
12 82 86
12 83 85
12 84 84
13 13 154
13 14 153
13 15 152
13 16 151
13 17 150
13 18 149
13 19 148
13 20 147
13 21 146
13 22 145
13 23 144
13 24 143
13 25 142
13 26 141
13 27 140
13 28 139
13 29 138
13 30 137
13 31 136
13 32 135
13 33 134
13 34 133
13 35 132
13 36 131
13 37 130
13 38 129
13 39 128
13 40 127
13 41 126
13 42 125
13 43 124
13 44 123
13 45 122
13 46 121
13 47 120
13 48 119
13 49 118
13 50 117
13 51 116
13 52 115
13 53 114
13 54 113
13 55 112
13 56 111
13 57 110
13 58 109
13 59 108
13 60 107
13 61 106
13 62 105
13 63 104
13 64 103
13 65 102
13 66 101
13 67 100
13 68 99
13 69 98
13 70 97
13 71 96
13 72 95
13 73 94
13 74 93
13 75 92
13 76 91
13 77 90
13 78 89
13 79 88
13 80 87
13 81 86
13 82 85
13 83 84
14 14 152
14 15 151
14 16 150
14 17 149
14 18 148
14 19 147
14 20 146
14 21 145
14 22 144
14 23 143
14 24 142
14 25 141
14 26 140
14 27 139
14 28 138
14 29 137
14 30 136
14 31 135
14 32 134
14 33 133
14 34 132
14 35 131
14 36 130
14 37 129
14 38 128
14 39 127
14 40 126
14 41 125
14 42 124
14 43 123
14 44 122
14 45 121
14 46 120
14 47 119
14 48 118
14 49 117
14 50 116
14 51 115
14 52 114
14 53 113
14 54 112
14 55 111
14 56 110
14 57 109
14 58 108
14 59 107
14 60 106
14 61 105
14 62 104
14 63 103
14 64 102
14 65 101
14 66 100
14 67 99
14 68 98
14 69 97
14 70 96
14 71 95
14 72 94
14 73 93
14 74 92
14 75 91
14 76 90
14 77 89
14 78 88
14 79 87
14 80 86
14 81 85
14 82 84
14 83 83
15 15 150
15 16 149
15 17 148
15 18 147
15 19 146
15 20 145
15 21 144
15 22 143
15 23 142
15 24 141
15 25 140
15 26 139
15 27 138
15 28 137
15 29 136
15 30 135
15 31 134
15 32 133
15 33 132
15 34 131
15 35 130
15 36 129
15 37 128
15 38 127
15 39 126
15 40 125
15 41 124
15 42 123
15 43 122
15 44 121
15 45 120
15 46 119
15 47 118
15 48 117
15 49 116
15 50 115
15 51 114
15 52 113
15 53 112
15 54 111
15 55 110
15 56 109
15 57 108
15 58 107
15 59 106
15 60 105
15 61 104
15 62 103
15 63 102
15 64 101
15 65 100
15 66 99
15 67 98
15 68 97
15 69 96
15 70 95
15 71 94
15 72 93
15 73 92
15 74 91
15 75 90
15 76 89
15 77 88
15 78 87
15 79 86
15 80 85
15 81 84
15 82 83
16 16 148
16 17 147
16 18 146
16 19 145
16 20 144
16 21 143
16 22 142
16 23 141
16 24 140
16 25 139
16 26 138
16 27 137
16 28 136
16 29 135
16 30 134
16 31 133
16 32 132
16 33 131
16 34 130
16 35 129
16 36 128
16 37 127
16 38 126
16 39 125
16 40 124
16 41 123
16 42 122
16 43 121
16 44 120
16 45 119
16 46 118
16 47 117
16 48 116
16 49 115
16 50 114
16 51 113
16 52 112
16 53 111
16 54 110
16 55 109
16 56 108
16 57 107
16 58 106
16 59 105
16 60 104
16 61 103
16 62 102
16 63 101
16 64 100
16 65 99
16 66 98
16 67 97
16 68 96
16 69 95
16 70 94
16 71 93
16 72 92
16 73 91
16 74 90
16 75 89
16 76 88
16 77 87
16 78 86
16 79 85
16 80 84
16 81 83
16 82 82
17 17 146
17 18 145
17 19 144
17 20 143
17 21 142
17 22 141
17 23 140
17 24 139
17 25 138
17 26 137
17 27 136
17 28 135
17 29 134
17 30 133
17 31 132
17 32 131
17 33 130
17 34 129
17 35 128
17 36 127
17 37 126
17 38 125
17 39 124
17 40 123
17 41 122
17 42 121
17 43 120
17 44 119
17 45 118
17 46 117
17 47 116
17 48 115
17 49 114
17 50 113
17 51 112
17 52 111
17 53 110
17 54 109
17 55 108
17 56 107
17 57 106
17 58 105
17 59 104
17 60 103
17 61 102
17 62 101
17 63 100
17 64 99
17 65 98
17 66 97
17 67 96
17 68 95
17 69 94
17 70 93
17 71 92
17 72 91
17 73 90
17 74 89
17 75 88
17 76 87
17 77 86
17 78 85
17 79 84
17 80 83
17 81 82
18 18 144
18 19 143
18 20 142
18 21 141
18 22 140
18 23 139
18 24 138
18 25 137
18 26 136
18 27 135
18 28 134
18 29 133
18 30 132
18 31 131
18 32 130
18 33 129
18 34 128
18 35 127
18 36 126
18 37 125
18 38 124
18 39 123
18 40 122
18 41 121
18 42 120
18 43 119
18 44 118
18 45 117
18 46 116
18 47 115
18 48 114
18 49 113
18 50 112
18 51 111
18 52 110
18 53 109
18 54 108
18 55 107
18 56 106
18 57 105
18 58 104
18 59 103
18 60 102
18 61 101
18 62 100
18 63 99
18 64 98
18 65 97
18 66 96
18 67 95
18 68 94
18 69 93
18 70 92
18 71 91
18 72 90
18 73 89
18 74 88
18 75 87
18 76 86
18 77 85
18 78 84
18 79 83
18 80 82
18 81 81
19 19 142
19 20 141
19 21 140
19 22 139
19 23 138
19 24 137
19 25 136
19 26 135
19 27 134
19 28 133
19 29 132
19 30 131
19 31 130
19 32 129
19 33 128
19 34 127
19 35 126
19 36 125
19 37 124
19 38 123
19 39 122
19 40 121
19 41 120
19 42 119
19 43 118
19 44 117
19 45 116
19 46 115
19 47 114
19 48 113
19 49 112
19 50 111
19 51 110
19 52 109
19 53 108
19 54 107
19 55 106
19 56 105
19 57 104
19 58 103
19 59 102
19 60 101
19 61 100
19 62 99
19 63 98
19 64 97
19 65 96
19 66 95
19 67 94
19 68 93
19 69 92
19 70 91
19 71 90
19 72 89
19 73 88
19 74 87
19 75 86
19 76 85
19 77 84
19 78 83
19 79 82
19 80 81
20 20 140
20 21 139
20 22 138
20 23 137
20 24 136
20 25 135
20 26 134
20 27 133
20 28 132
20 29 131
20 30 130
20 31 129
20 32 128
20 33 127
20 34 126
20 35 125
20 36 124
20 37 123
20 38 122
20 39 121
20 40 120
20 41 119
20 42 118
20 43 117
20 44 116
20 45 115
20 46 114
20 47 113
20 48 112
20 49 111
20 50 110
20 51 109
20 52 108
20 53 107
20 54 106
20 55 105
20 56 104
20 57 103
20 58 102
20 59 101
20 60 100
20 61 99
20 62 98
20 63 97
20 64 96
20 65 95
20 66 94
20 67 93
20 68 92
20 69 91
20 70 90
20 71 89
20 72 88
20 73 87
20 74 86
20 75 85
20 76 84
20 77 83
20 78 82
20 79 81
20 80 80
21 21 138
21 22 137
21 23 136
21 24 135
21 25 134
21 26 133
21 27 132
21 28 131
21 29 130
21 30 129
21 31 128
21 32 127
21 33 126
21 34 125
21 35 124
21 36 123
21 37 122
21 38 121
21 39 120
21 40 119
21 41 118
21 42 117
21 43 116
21 44 115
21 45 114
21 46 113
21 47 112
21 48 111
21 49 110
21 50 109
21 51 108
21 52 107
21 53 106
21 54 105
21 55 104
21 56 103
21 57 102
21 58 101
21 59 100
21 60 99
21 61 98
21 62 97
21 63 96
21 64 95
21 65 94
21 66 93
21 67 92
21 68 91
21 69 90
21 70 89
21 71 88
21 72 87
21 73 86
21 74 85
21 75 84
21 76 83
21 77 82
21 78 81
21 79 80
22 22 136
22 23 135
22 24 134
22 25 133
22 26 132
22 27 131
22 28 130
22 29 129
22 30 128
22 31 127
22 32 126
22 33 125
22 34 124
22 35 123
22 36 122
22 37 121
22 38 120
22 39 119
22 40 118
22 41 117
22 42 116
22 43 115
22 44 114
22 45 113
22 46 112
22 47 111
22 48 110
22 49 109
22 50 108
22 51 107
22 52 106
22 53 105
22 54 104
22 55 103
22 56 102
22 57 101
22 58 100
22 59 99
22 60 98
22 61 97
22 62 96
22 63 95
22 64 94
22 65 93
22 66 92
22 67 91
22 68 90
22 69 89
22 70 88
22 71 87
22 72 86
22 73 85
22 74 84
22 75 83
22 76 82
22 77 81
22 78 80
22 79 79
23 23 134
23 24 133
23 25 132
23 26 131
23 27 130
23 28 129
23 29 128
23 30 127
23 31 126
23 32 125
23 33 124
23 34 123
23 35 122
23 36 121
23 37 120
23 38 119
23 39 118
23 40 117
23 41 116
23 42 115
23 43 114
23 44 113
23 45 112
23 46 111
23 47 110
23 48 109
23 49 108
23 50 107
23 51 106
23 52 105
23 53 104
23 54 103
23 55 102
23 56 101
23 57 100
23 58 99
23 59 98
23 60 97
23 61 96
23 62 95
23 63 94
23 64 93
23 65 92
23 66 91
23 67 90
23 68 89
23 69 88
23 70 87
23 71 86
23 72 85
23 73 84
23 74 83
23 75 82
23 76 81
23 77 80
23 78 79
24 24 132
24 25 131
24 26 130
24 27 129
24 28 128
24 29 127
24 30 126
24 31 125
24 32 124
24 33 123
24 34 122
24 35 121
24 36 120
24 37 119
24 38 118
24 39 117
24 40 116
24 41 115
24 42 114
24 43 113
24 44 112
24 45 111
24 46 110
24 47 109
24 48 108
24 49 107
24 50 106
24 51 105
24 52 104
24 53 103
24 54 102
24 55 101
24 56 100
24 57 99
24 58 98
24 59 97
24 60 96
24 61 95
24 62 94
24 63 93
24 64 92
24 65 91
24 66 90
24 67 89
24 68 88
24 69 87
24 70 86
24 71 85
24 72 84
24 73 83
24 74 82
24 75 81
24 76 80
24 77 79
24 78 78
25 25 130
25 26 129
25 27 128
25 28 127
25 29 126
25 30 125
25 31 124
25 32 123
25 33 122
25 34 121
25 35 120
25 36 119
25 37 118
25 38 117
25 39 116
25 40 115
25 41 114
25 42 113
25 43 112
25 44 111
25 45 110
25 46 109
25 47 108
25 48 107
25 49 106
25 50 105
25 51 104
25 52 103
25 53 102
25 54 101
25 55 100
25 56 99
25 57 98
25 58 97
25 59 96
25 60 95
25 61 94
25 62 93
25 63 92
25 64 91
25 65 90
25 66 89
25 67 88
25 68 87
25 69 86
25 70 85
25 71 84
25 72 83
25 73 82
25 74 81
25 75 80
25 76 79
25 77 78
26 26 128
26 27 127
26 28 126
26 29 125
26 30 124
26 31 123
26 32 122
26 33 121
26 34 120
26 35 119
26 36 118
26 37 117
26 38 116
26 39 115
26 40 114
26 41 113
26 42 112
26 43 111
26 44 110
26 45 109
26 46 108
26 47 107
26 48 106
26 49 105
26 50 104
26 51 103
26 52 102
26 53 101
26 54 100
26 55 99
26 56 98
26 57 97
26 58 96
26 59 95
26 60 94
26 61 93
26 62 92
26 63 91
26 64 90
26 65 89
26 66 88
26 67 87
26 68 86
26 69 85
26 70 84
26 71 83
26 72 82
26 73 81
26 74 80
26 75 79
26 76 78
26 77 77
27 27 126
27 28 125
27 29 124
27 30 123
27 31 122
27 32 121
27 33 120
27 34 119
27 35 118
27 36 117
27 37 116
27 38 115
27 39 114
27 40 113
27 41 112
27 42 111
27 43 110
27 44 109
27 45 108
27 46 107
27 47 106
27 48 105
27 49 104
27 50 103
27 51 102
27 52 101
27 53 100
27 54 99
27 55 98
27 56 97
27 57 96
27 58 95
27 59 94
27 60 93
27 61 92
27 62 91
27 63 90
27 64 89
27 65 88
27 66 87
27 67 86
27 68 85
27 69 84
27 70 83
27 71 82
27 72 81
27 73 80
27 74 79
27 75 78
27 76 77
28 28 124
28 29 123
28 30 122
28 31 121
28 32 120
28 33 119
28 34 118
28 35 117
28 36 116
28 37 115
28 38 114
28 39 113
28 40 112
28 41 111
28 42 110
28 43 109
28 44 108
28 45 107
28 46 106
28 47 105
28 48 104
28 49 103
28 50 102
28 51 101
28 52 100
28 53 99
28 54 98
28 55 97
28 56 96
28 57 95
28 58 94
28 59 93
28 60 92
28 61 91
28 62 90
28 63 89
28 64 88
28 65 87
28 66 86
28 67 85
28 68 84
28 69 83
28 70 82
28 71 81
28 72 80
28 73 79
28 74 78
28 75 77
28 76 76
29 29 122
29 30 121
29 31 120
29 32 119
29 33 118
29 34 117
29 35 116
29 36 115
29 37 114
29 38 113
29 39 112
29 40 111
29 41 110
29 42 109
29 43 108
29 44 107
29 45 106
29 46 105
29 47 104
29 48 103
29 49 102
29 50 101
29 51 100
29 52 99
29 53 98
29 54 97
29 55 96
29 56 95
29 57 94
29 58 93
29 59 92
29 60 91
29 61 90
29 62 89
29 63 88
29 64 87
29 65 86
29 66 85
29 67 84
29 68 83
29 69 82
29 70 81
29 71 80
29 72 79
29 73 78
29 74 77
29 75 76
30 30 120
30 31 119
30 32 118
30 33 117
30 34 116
30 35 115
30 36 114
30 37 113
30 38 112
30 39 111
30 40 110
30 41 109
30 42 108
30 43 107
30 44 106
30 45 105
30 46 104
30 47 103
30 48 102
30 49 101
30 50 100
30 51 99
30 52 98
30 53 97
30 54 96
30 55 95
30 56 94
30 57 93
30 58 92
30 59 91
30 60 90
30 61 89
30 62 88
30 63 87
30 64 86
30 65 85
30 66 84
30 67 83
30 68 82
30 69 81
30 70 80
30 71 79
30 72 78
30 73 77
30 74 76
30 75 75
31 31 118
31 32 117
31 33 116
31 34 115
31 35 114
31 36 113
31 37 112
31 38 111
31 39 110
31 40 109
31 41 108
31 42 107
31 43 106
31 44 105
31 45 104
31 46 103
31 47 102
31 48 101
31 49 100
31 50 99
31 51 98
31 52 97
31 53 96
31 54 95
31 55 94
31 56 93
31 57 92
31 58 91
31 59 90
31 60 89
31 61 88
31 62 87
31 63 86
31 64 85
31 65 84
31 66 83
31 67 82
31 68 81
31 69 80
31 70 79
31 71 78
31 72 77
31 73 76
31 74 75
32 32 116
32 33 115
32 34 114
32 35 113
32 36 112
32 37 111
32 38 110
32 39 109
32 40 108
32 41 107
32 42 106
32 43 105
32 44 104
32 45 103
32 46 102
32 47 101
32 48 100
32 49 99
32 50 98
32 51 97
32 52 96
32 53 95
32 54 94
32 55 93
32 56 92
32 57 91
32 58 90
32 59 89
32 60 88
32 61 87
32 62 86
32 63 85
32 64 84
32 65 83
32 66 82
32 67 81
32 68 80
32 69 79
32 70 78
32 71 77
32 72 76
32 73 75
32 74 74
33 33 114
33 34 113
33 35 112
33 36 111
33 37 110
33 38 109
33 39 108
33 40 107
33 41 106
33 42 105
33 43 104
33 44 103
33 45 102
33 46 101
33 47 100
33 48 99
33 49 98
33 50 97
33 51 96
33 52 95
33 53 94
33 54 93
33 55 92
33 56 91
33 57 90
33 58 89
33 59 88
33 60 87
33 61 86
33 62 85
33 63 84
33 64 83
33 65 82
33 66 81
33 67 80
33 68 79
33 69 78
33 70 77
33 71 76
33 72 75
33 73 74
34 34 112
34 35 111
34 36 110
34 37 109
34 38 108
34 39 107
34 40 106
34 41 105
34 42 104
34 43 103
34 44 102
34 45 101
34 46 100
34 47 99
34 48 98
34 49 97
34 50 96
34 51 95
34 52 94
34 53 93
34 54 92
34 55 91
34 56 90
34 57 89
34 58 88
34 59 87
34 60 86
34 61 85
34 62 84
34 63 83
34 64 82
34 65 81
34 66 80
34 67 79
34 68 78
34 69 77
34 70 76
34 71 75
34 72 74
34 73 73
35 35 110
35 36 109
35 37 108
35 38 107
35 39 106
35 40 105
35 41 104
35 42 103
35 43 102
35 44 101
35 45 100
35 46 99
35 47 98
35 48 97
35 49 96
35 50 95
35 51 94
35 52 93
35 53 92
35 54 91
35 55 90
35 56 89
35 57 88
35 58 87
35 59 86
35 60 85
35 61 84
35 62 83
35 63 82
35 64 81
35 65 80
35 66 79
35 67 78
35 68 77
35 69 76
35 70 75
35 71 74
35 72 73
36 36 108
36 37 107
36 38 106
36 39 105
36 40 104
36 41 103
36 42 102
36 43 101
36 44 100
36 45 99
36 46 98
36 47 97
36 48 96
36 49 95
36 50 94
36 51 93
36 52 92
36 53 91
36 54 90
36 55 89
36 56 88
36 57 87
36 58 86
36 59 85
36 60 84
36 61 83
36 62 82
36 63 81
36 64 80
36 65 79
36 66 78
36 67 77
36 68 76
36 69 75
36 70 74
36 71 73
36 72 72
37 37 106
37 38 105
37 39 104
37 40 103
37 41 102
37 42 101
37 43 100
37 44 99
37 45 98
37 46 97
37 47 96
37 48 95
37 49 94
37 50 93
37 51 92
37 52 91
37 53 90
37 54 89
37 55 88
37 56 87
37 57 86
37 58 85
37 59 84
37 60 83
37 61 82
37 62 81
37 63 80
37 64 79
37 65 78
37 66 77
37 67 76
37 68 75
37 69 74
37 70 73
37 71 72
38 38 104
38 39 103
38 40 102
38 41 101
38 42 100
38 43 99
38 44 98
38 45 97
38 46 96
38 47 95
38 48 94
38 49 93
38 50 92
38 51 91
38 52 90
38 53 89
38 54 88
38 55 87
38 56 86
38 57 85
38 58 84
38 59 83
38 60 82
38 61 81
38 62 80
38 63 79
38 64 78
38 65 77
38 66 76
38 67 75
38 68 74
38 69 73
38 70 72
38 71 71
39 39 102
39 40 101
39 41 100
39 42 99
39 43 98
39 44 97
39 45 96
39 46 95
39 47 94
39 48 93
39 49 92
39 50 91
39 51 90
39 52 89
39 53 88
39 54 87
39 55 86
39 56 85
39 57 84
39 58 83
39 59 82
39 60 81
39 61 80
39 62 79
39 63 78
39 64 77
39 65 76
39 66 75
39 67 74
39 68 73
39 69 72
39 70 71
40 40 100
40 41 99
40 42 98
40 43 97
40 44 96
40 45 95
40 46 94
40 47 93
40 48 92
40 49 91
40 50 90
40 51 89
40 52 88
40 53 87
40 54 86
40 55 85
40 56 84
40 57 83
40 58 82
40 59 81
40 60 80
40 61 79
40 62 78
40 63 77
40 64 76
40 65 75
40 66 74
40 67 73
40 68 72
40 69 71
40 70 70
41 41 98
41 42 97
41 43 96
41 44 95
41 45 94
41 46 93
41 47 92
41 48 91
41 49 90
41 50 89
41 51 88
41 52 87
41 53 86
41 54 85
41 55 84
41 56 83
41 57 82
41 58 81
41 59 80
41 60 79
41 61 78
41 62 77
41 63 76
41 64 75
41 65 74
41 66 73
41 67 72
41 68 71
41 69 70
42 42 96
42 43 95
42 44 94
42 45 93
42 46 92
42 47 91
42 48 90
42 49 89
42 50 88
42 51 87
42 52 86
42 53 85
42 54 84
42 55 83
42 56 82
42 57 81
42 58 80
42 59 79
42 60 78
42 61 77
42 62 76
42 63 75
42 64 74
42 65 73
42 66 72
42 67 71
42 68 70
42 69 69
43 43 94
43 44 93
43 45 92
43 46 91
43 47 90
43 48 89
43 49 88
43 50 87
43 51 86
43 52 85
43 53 84
43 54 83
43 55 82
43 56 81
43 57 80
43 58 79
43 59 78
43 60 77
43 61 76
43 62 75
43 63 74
43 64 73
43 65 72
43 66 71
43 67 70
43 68 69
44 44 92
44 45 91
44 46 90
44 47 89
44 48 88
44 49 87
44 50 86
44 51 85
44 52 84
44 53 83
44 54 82
44 55 81
44 56 80
44 57 79
44 58 78
44 59 77
44 60 76
44 61 75
44 62 74
44 63 73
44 64 72
44 65 71
44 66 70
44 67 69
44 68 68
45 45 90
45 46 89
45 47 88
45 48 87
45 49 86
45 50 85
45 51 84
45 52 83
45 53 82
45 54 81
45 55 80
45 56 79
45 57 78
45 58 77
45 59 76
45 60 75
45 61 74
45 62 73
45 63 72
45 64 71
45 65 70
45 66 69
45 67 68
46 46 88
46 47 87
46 48 86
46 49 85
46 50 84
46 51 83
46 52 82
46 53 81
46 54 80
46 55 79
46 56 78
46 57 77
46 58 76
46 59 75
46 60 74
46 61 73
46 62 72
46 63 71
46 64 70
46 65 69
46 66 68
46 67 67
47 47 86
47 48 85
47 49 84
47 50 83
47 51 82
47 52 81
47 53 80
47 54 79
47 55 78
47 56 77
47 57 76
47 58 75
47 59 74
47 60 73
47 61 72
47 62 71
47 63 70
47 64 69
47 65 68
47 66 67
48 48 84
48 49 83
48 50 82
48 51 81
48 52 80
48 53 79
48 54 78
48 55 77
48 56 76
48 57 75
48 58 74
48 59 73
48 60 72
48 61 71
48 62 70
48 63 69
48 64 68
48 65 67
48 66 66
49 49 82
49 50 81
49 51 80
49 52 79
49 53 78
49 54 77
49 55 76
49 56 75
49 57 74
49 58 73
49 59 72
49 60 71
49 61 70
49 62 69
49 63 68
49 64 67
49 65 66
50 50 80
50 51 79
50 52 78
50 53 77
50 54 76
50 55 75
50 56 74
50 57 73
50 58 72
50 59 71
50 60 70
50 61 69
50 62 68
50 63 67
50 64 66
50 65 65
51 51 78
51 52 77
51 53 76
51 54 75
51 55 74
51 56 73
51 57 72
51 58 71
51 59 70
51 60 69
51 61 68
51 62 67
51 63 66
51 64 65
52 52 76
52 53 75
52 54 74
52 55 73
52 56 72
52 57 71
52 58 70
52 59 69
52 60 68
52 61 67
52 62 66
52 63 65
52 64 64
53 53 74
53 54 73
53 55 72
53 56 71
53 57 70
53 58 69
53 59 68
53 60 67
53 61 66
53 62 65
53 63 64
54 54 72
54 55 71
54 56 70
54 57 69
54 58 68
54 59 67
54 60 66
54 61 65
54 62 64
54 63 63
55 55 70
55 56 69
55 57 68
55 58 67
55 59 66
55 60 65
55 61 64
55 62 63
56 56 68
56 57 67
56 58 66
56 59 65
56 60 64
56 61 63
56 62 62
57 57 66
57 58 65
57 59 64
57 60 63
57 61 62
58 58 64
58 59 63
58 60 62
58 61 61
59 59 62
59 60 61
60 60 60
 
Now I get to extend a mea culpa. I screwed up in the second half of my post.


The sum of the angles of any triangle is 180 degrees.

Start with the first angle and call it x. Valid integer values for x range from 1 to 178. (Yes, some of these triangles will look very bizarre.)

Call the second angle y. For each value of x, valid values for y range from 1 to 179-x.

Call the third angle z. It's value is fixed to a single value by the values of x and y (z=180-x-y) so it doesn't add to the number of triangles.

So you get the following summation:

178
-----
\
/ (179-x)
-----
x=1

You can use the trick of the sum from 1 to n = n*(n+1)/2 to turn it into

178*179 - 178*(178+1)/2 = 15,931

Now to account for similar triangles. If all three angles have unique values there are 6 permutations of those values so the total would have to be divided by 6.

But what about the cases where the three angles don't have unique values. When two are the same, the third angle must have an even value so there are 178/2 = 89 of these. Since there are 3 permutations for the third angle we have 89*3 = 267 to add to the total. Finally, we have to add 2 more to account for the special case of 60/60/60 where all three angles are the same.

15,931+267+2 = 16,200

Divide this by six to remove the similarities and you get 2,700.

See. Let me fix it and I give you a complete solution. :)
 
You must log in or register to reply here.
bread's done
Back
Top