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 3 equal products (Posted on 2018-11-21)
Lets take a number 12 as an example . It can be partitoned into three integer summands in 12 ways:
12=1+1+10
12=1+2+9
12=1+3+8
12=1+4+7
12=1+5+6
12=2+2+8
12=2+3+7
12=2+4+6
12=2+5+5
12=3+3+6
12=3+4+5
12=4+4+4

Multiplying the 3 members of each partitions results in 12 distinct numbers: 10,18,...60,64.

On the other hand the same treatment applied to number 13 produces a pair of equal results: 13=1+6+6=2+2+9 and 1*6*6=2*2*9=36 (a well known problem of children's ages).

Find the smallest number which has 3 distinct partitions into 3 parts, each of them with the same product.

Bonus: list all numbers below 1000 boasting this feature.

 No Solution Yet Submitted by Ady TZIDON No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(2): computer solution and partial investigation of the bonus - ctd. | Comment 3 of 10 |
(In reply to re: computer solution and partial investigation of the bonus - ctd. by Charlie)

224 4 201600
20 84 120
21 75 128
24 60 140
32 42 150

224 3 238464
24 92 108
27 69 128
32 54 138

224 4 266112
28 88 108
32 66 126
33 63 128
36 56 132

224 3 280800
30 90 104
32 75 117
40 54 130

224 3 293760
32 90 102
36 68 120
45 51 128

224 3 325584
38 84 102
42 68 114
48 57 119

224 3 333450
39 90 95
45 65 114
50 57 117

224 4 356400
44 90 90
45 80 99
50 66 108
54 60 110

224 3 374400
50 78 96
52 72 100
60 60 104

225 3 12540
1 110 114
2 33 190
6 10 209

225 3 71280
6 99 120
8 55 162
12 33 180

225 3 95760
9 76 140
10 63 152
14 40 171

225 3 114240
10 96 119
14 51 160
17 40 168

225 3 120120
11 84 130
12 70 143
14 55 156

225 4 154560
14 96 115
16 69 140
23 42 160
24 40 161

225 3 158760
15 84 126
18 60 147
28 35 162

225 4 191520
18 95 112
19 80 126
20 72 133
28 45 152

225 3 196080
19 86 120
20 76 129
30 43 152

225 3 210000
20 100 105
25 60 140
35 40 150

225 4 240240
24 91 110
28 65 132
33 52 140
40 42 143

225 3 251160
26 84 115
35 52 138
39 46 140

225 3 285120
30 96 99
33 72 120
45 48 132

225 3 295680
32 88 105
33 80 112
42 55 128

225 3 336960
39 90 96
40 81 104
48 60 117

225 3 351120
42 88 95
44 76 105
55 56 114

226 3 30240
3 63 160
4 42 180
5 32 189

226 3 37026
3 102 121
6 33 187
11 17 198

226 3 43200
4 72 150
6 40 180
9 25 192

226 3 52416
6 52 168
8 36 182
13 21 192

226 3 54720
8 38 180
12 24 190
15 19 192

226 3 113400
10 90 126
14 50 162
24 27 175

226 3 137088
12 102 112
14 68 144
24 34 168

226 3 141120
14 72 140
15 64 147
28 30 168

226 3 176256
16 102 108
17 81 128
18 72 136

226 3 194040
18 98 110
24 55 147
30 42 154

226 3 198360
19 87 120
24 57 145
29 45 152

226 3 198450
21 70 135
25 54 147
27 49 150

226 3 216216
21 88 117
22 78 126
27 56 143

226 3 241920
24 90 112
32 54 140
40 42 144

226 3 251100
25 93 108
27 75 124
31 60 135

226 3 274176
28 96 102
32 68 126
42 48 136

226 3 335160
38 90 98
42 70 114
49 57 120

226 3 336960
40 78 108
45 64 117
52 54 120

227 3 66528
6 77 144
9 42 176
16 22 189

227 3 91800
9 68 150
12 45 170
17 30 180

227 3 110880
14 48 165
15 44 168
21 30 176

227 3 131040
13 70 144
15 56 156
20 39 168

227 3 139104
14 69 144
18 48 161
23 36 168

227 3 148005
13 99 115
15 69 143
23 39 165

227 3 158400
15 80 132
22 45 160
30 32 165

227 3 168480
15 104 108
16 81 130
18 65 144

227 3 186048
17 96 114
19 72 136
24 51 152

227 3 208800
20 87 120
25 58 144
29 48 150

227 3 239904
24 84 119
28 63 136
34 49 144

227 3 247104
24 99 104
36 48 143
39 44 144

227 3 277200
28 99 100
30 77 120
35 60 132

227 3 327600
36 91 100
40 70 117
42 65 120

227 3 372600
45 90 92
46 81 100
50 69 108

228 3 11760
1 80 147
2 30 196
4 14 210

228 3 37422
3 99 126
6 33 189
9 21 198

228 3 84000
7 96 125
8 70 150
10 50 168

228 4 118800
10 108 110
12 66 150
15 48 165
25 27 176

228 3 148512
13 96 119
14 78 136
26 34 168

228 3 184800
22 56 150
24 50 154
33 35 160

228 3 192192
21 64 143
26 48 154
28 44 156

228 3 210210
21 77 130
26 55 147
35 39 154

228 3 268800
28 80 120
30 70 128
40 48 140

228 3 369600
44 84 100
48 70 110
50 66 112

229 3 24840
2 92 135
3 46 180
10 12 207

229 3 39600
4 60 165
5 44 180
11 18 200

229 3 72072
6 91 132
7 66 156
11 36 182

229 3 108000
9 100 120
10 75 144
24 25 180

229 3 123200
14 55 160
22 32 175
25 28 176

229 3 134064
12 84 133
14 63 152
19 42 168

229 3 144144
13 84 132
14 72 143
22 39 168

229 3 146880
16 60 153
18 51 160
27 32 170

229 4 151200
14 80 135
15 70 144
16 63 150
25 36 168

229 3 171360
15 102 112
17 72 140
20 56 153

229 3 181440
16 105 108
27 42 160
32 35 162

229 3 187200
20 65 144
25 48 156
30 39 160

229 3 199584
18 99 112
22 63 144
27 48 154

229 3 209475
19 105 105
21 75 133
25 57 147

229 3 243432
23 98 108
28 63 138
36 46 147

229 3 247104
24 88 117
32 54 143
33 52 144

229 3 252000
24 100 105
25 84 120
35 50 144

229 3 254475
25 87 117
29 65 135
39 45 145

229 3 259200
25 96 108
30 64 135
40 45 144

229 3 294840
30 91 108
36 63 130
42 52 135

229 3 310464
32 98 99
33 84 112
48 49 132

229 3 352800
40 84 105
42 75 112
49 60 120

229 3 354960
40 87 102
45 68 116
51 58 120

229 3 393120
48 90 91
52 72 105
56 65 108

229 3 396900
49 90 90
50 81 98
54 70 105

229 3 400400
50 88 91
52 77 100
55 70 104

229 3 415800
55 84 90
60 70 99
63 66 100

230 3 20160
2 60 168
3 35 192
8 12 210

230 3 75240
6 110 114
8 57 165
18 22 190

230 3 97920
8 102 120
12 48 170
16 34 180

230 3 161280
14 96 120
16 70 144
30 32 168

230 3 242190
23 90 117
26 69 135
27 65 138

230 3 297000
30 90 110
33 72 125
40 55 135

230 3 299520
30 96 104
32 78 120
36 64 130

230 3 299880
30 98 102
34 70 126
45 49 136

230 3 328320
36 80 114
38 72 120
45 57 128

230 3 332640
35 96 99
36 84 110
44 60 126

231 3 62400
5 96 130
6 65 160
16 20 195

231 3 85008
7 92 132
8 69 154
14 33 184

231 4 96096
8 91 132
11 52 168
13 42 176
16 33 182

231 3 106920
9 90 132
12 54 165
18 33 180

231 3 120384
11 76 144
16 44 171
19 36 176

231 3 132000
11 100 120
16 50 165
25 30 176

231 3 132825
11 105 115
15 55 161
23 33 175

231 3 157080
14 85 132
17 60 154
28 33 170

231 3 163200
15 80 136
17 64 150
20 51 160

231 3 166320
15 84 132
24 42 165
30 33 168

231 3 188496
17 88 126
18 77 136
22 56 153

231 3 200640
19 80 132
24 55 152
33 38 160

231 3 205200
20 76 135
24 57 150
25 54 152

231 3 218400
20 91 120
21 80 130
35 40 156

231 3 231000
21 100 110
22 84 125
25 66 140

231 3 240240
22 104 105
24 77 130
28 60 143

231 3 244200
25 74 132
33 50 148
37 44 150

231 3 256608
24 99 108
27 72 132
33 54 144

231 3 265200
25 102 104
26 85 120
30 65 136

231 3 267960
29 70 132
33 58 140
42 44 145

231 3 308880
33 78 120
39 60 132
44 52 135

231 3 316008
33 84 114
42 57 132
44 54 133

231 3 386400
46 80 105
50 69 112
56 60 115

232 3 47628
4 81 147
7 36 189
9 27 196

232 3 56430
6 55 171
9 33 190
15 19 198

232 3 63360
5 99 128
6 66 160
8 44 180

232 3 75600
6 100 126
9 48 175
10 42 180

232 3 99900
9 75 148
15 37 180
20 27 185

232 3 110250
9 98 125
10 75 147
15 42 175

232 3 185328
16 99 117
22 54 156
26 44 162

232 3 206720
19 85 128
20 76 136
34 38 160

232 3 232848
21 99 112
22 84 126
36 42 154

232 3 240768
22 96 114
24 76 132
36 44 152

232 3 242550
22 105 105
30 55 147
33 49 150

232 3 263250
25 90 117
26 81 125
27 75 130

232 3 275184
26 98 108
28 78 126
39 49 144

232 3 287280
28 90 114
30 76 126
38 54 140

232 3 291200
28 100 104
32 70 130
40 52 140

232 3 308448
32 81 119
34 72 126
42 54 136

232 3 325584
34 84 114
38 68 126
48 51 133

232 3 326700
33 99 100
36 75 121
45 55 132

232 3 345600
36 96 100
40 72 120
50 54 128

233 3 72576
7 64 162
12 32 189
14 27 192

233 3 118800
10 88 135
11 72 150
20 33 180

233 3 122760
10 99 124
12 66 155
22 31 180

233 3 126000
15 50 168
18 40 175
25 28 180

233 3 133056
11 96 126
12 77 144
21 36 176

233 3 165312
14 96 123
21 48 164
24 41 168

233 3 205632
18 96 119
21 68 144
24 56 153

233 3 207000
18 100 115
20 75 138
23 60 150

233 3 212800
20 80 133
25 56 152
35 38 160

233 3 224640
20 96 117
24 65 144
32 45 156

233 3 235620
21 102 110
22 85 126
34 45 154

233 3 244800
25 72 136
32 51 150
40 40 153

233 3 257400
26 75 132
30 60 143
39 44 150

233 3 264600
28 70 135
30 63 140
36 50 147

233 3 284256
27 94 112
36 56 141
42 47 144

233 3 285120
27 96 110
32 66 135
44 45 144

233 3 334152
34 91 108
36 78 119
39 68 126

233 3 383040
42 95 96
45 76 112
56 57 120

233 3 415800
50 84 99
55 70 108
60 63 110

233 3 421200
52 81 100
54 75 104
60 65 108

233 3 432432
54 88 91
56 78 99
63 66 104

DefDbl A-Z
Dim crlf\$, dr

Form1.Visible = True

pi = Atn(1) * 4
dr = pi / 180

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For n = 6 To 243
ReDim prd(999999, 5, 3)
lg = 0
For a = 1 To n / 3
For b = a To (n - a) / 2
c = n - a - b
If c > 0 Then
p = a * b * c
If p > lg Then lg = p
prd(p, 0, 0) = prd(p, 0, 0) + 1
prd(p, prd(p, 0, 0), 1) = a
prd(p, prd(p, 0, 0), 2) = b
prd(p, prd(p, 0, 0), 3) = c
End If
Next
Next
For i = 1 To lg
If prd(i, 0, 0) >= 3 Then
Text1.Text = Text1.Text & n & Str(prd(i, 0, 0)) & Str(i) & crlf
For j = 1 To prd(i, 0, 0)
Text1.Text = Text1.Text & Str(prd(i, j, 1)) & Str(prd(i, j, 2)) & Str(prd(i, j, 3)) & crlf
Next
Text1.Text = Text1.Text & crlf
End If
Next
DoEvents
Next

Text1.Text = Text1.Text & crlf & tot & " done"

End Sub

 Posted by Charlie on 2018-11-21 12:14:33

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