{
  "id": 312702,
  "title": "Digits sum analysis for 3, 4, 5 digits  ",
  "url": "/competitions/ultra-mnist/discussion/312702",
  "author_name": "",
  "post_date": "2022-03-13T16:05:48.552833900Z",
  "votes": 11,
  "comment_count": 1,
  "views": 0,
  "content": "<p>Here are a table that give how many way to add up 3, 4 and 5 digits to a given<br>\ndigit_sum.<br>\n<strong>Note</strong>: permutation are not counted (but you can compute them using factorial)</p>\n<table>\n<thead>\n<tr>\n<th><em>digit_sum</em></th>\n<th><em>x3</em></th>\n<th><em>x4</em></th>\n<th><em>x5</em></th>\n<th><em>row_sum</em></th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>0</td>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td><strong>3</strong></td>\n</tr>\n<tr>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td><strong>3</strong></td>\n</tr>\n<tr>\n<td>2</td>\n<td>2</td>\n<td>2</td>\n<td>2</td>\n<td><strong>6</strong></td>\n</tr>\n<tr>\n<td>3</td>\n<td>3</td>\n<td>3</td>\n<td>3</td>\n<td><strong>9</strong></td>\n</tr>\n<tr>\n<td>4</td>\n<td>4</td>\n<td>5</td>\n<td>5</td>\n<td><strong>14</strong></td>\n</tr>\n<tr>\n<td>5</td>\n<td>5</td>\n<td>6</td>\n<td>7</td>\n<td><strong>18</strong></td>\n</tr>\n<tr>\n<td>6</td>\n<td>7</td>\n<td>9</td>\n<td>10</td>\n<td><strong>26</strong></td>\n</tr>\n<tr>\n<td>7</td>\n<td>8</td>\n<td>11</td>\n<td>13</td>\n<td><strong>32</strong></td>\n</tr>\n<tr>\n<td>8</td>\n<td>10</td>\n<td>15</td>\n<td>18</td>\n<td><strong>43</strong></td>\n</tr>\n<tr>\n<td>9</td>\n<td>12</td>\n<td>18</td>\n<td>23</td>\n<td><strong>53</strong></td>\n</tr>\n<tr>\n<td>10</td>\n<td>13</td>\n<td>22</td>\n<td>29</td>\n<td><strong>64</strong></td>\n</tr>\n<tr>\n<td>11</td>\n<td>14</td>\n<td>25</td>\n<td>35</td>\n<td><strong>74</strong></td>\n</tr>\n<tr>\n<td>12</td>\n<td>15</td>\n<td>30</td>\n<td>43</td>\n<td><strong>88</strong></td>\n</tr>\n<tr>\n<td>13</td>\n<td>15</td>\n<td>32</td>\n<td>50</td>\n<td><strong>97</strong></td>\n</tr>\n<tr>\n<td>14</td>\n<td>15</td>\n<td>36</td>\n<td>58</td>\n<td><strong>109</strong></td>\n</tr>\n<tr>\n<td>15</td>\n<td>15</td>\n<td>38</td>\n<td>66</td>\n<td><strong>119</strong></td>\n</tr>\n<tr>\n<td>16</td>\n<td>14</td>\n<td>41</td>\n<td>74</td>\n<td><strong>129</strong></td>\n</tr>\n<tr>\n<td>17</td>\n<td>13</td>\n<td>41</td>\n<td>81</td>\n<td><strong>135</strong></td>\n</tr>\n<tr>\n<td>18</td>\n<td>12</td>\n<td>43</td>\n<td>88</td>\n<td><strong>143</strong></td>\n</tr>\n<tr>\n<td>19</td>\n<td>10</td>\n<td>41</td>\n<td>93</td>\n<td><strong>144</strong></td>\n</tr>\n<tr>\n<td>20</td>\n<td>8</td>\n<td>41</td>\n<td>98</td>\n<td><strong>147</strong></td>\n</tr>\n<tr>\n<td>21</td>\n<td>7</td>\n<td>38</td>\n<td>101</td>\n<td><strong>146</strong></td>\n</tr>\n<tr>\n<td>22</td>\n<td>5</td>\n<td>36</td>\n<td>102</td>\n<td><strong>143</strong></td>\n</tr>\n<tr>\n<td>23</td>\n<td>4</td>\n<td>32</td>\n<td>102</td>\n<td><strong>138</strong></td>\n</tr>\n<tr>\n<td>24</td>\n<td>3</td>\n<td>30</td>\n<td>101</td>\n<td><strong>134</strong></td>\n</tr>\n<tr>\n<td>25</td>\n<td>2</td>\n<td>25</td>\n<td>98</td>\n<td><strong>125</strong></td>\n</tr>\n<tr>\n<td>26</td>\n<td>1</td>\n<td>22</td>\n<td>93</td>\n<td><strong>116</strong></td>\n</tr>\n<tr>\n<td>27</td>\n<td>1</td>\n<td>18</td>\n<td>88</td>\n<td><strong>107</strong></td>\n</tr>\n<tr>\n<td><em>col_sum</em></td>\n<td><strong>220</strong></td>\n<td><strong>662</strong></td>\n<td><strong>1483</strong></td>\n<td><strong>2365</strong></td>\n</tr>\n</tbody>\n</table>\n<p>ie, for digit_sum=7 and 4 digits (a+b+c+d=7), there are 11 possibilites.</p>\n<p>Here the json file : <a href=\"https://pastebin.com/TTwrSu6f\" target=\"_blank\">https://pastebin.com/TTwrSu6f</a></p>\n<pre><code>&gt;&gt;&gt; json_data[4][7] \n[(1, 1, 2, 3),\n (0, 1, 1, 5),\n (0, 0, 0, 7),\n (0, 1, 3, 3),\n (1, 2, 2, 2),\n (0, 0, 2, 5),\n (1, 1, 1, 4),\n (0, 0, 1, 6),\n (0, 2, 2, 3),\n (0, 0, 3, 4),\n (0, 1, 2, 4)]\n</code></pre>",
  "messages": [
    {
      "id": "1721364",
      "postDate": "03/13/2022 16:05:48",
      "content": "<p>Here are a table that give how many way to add up 3, 4 and 5 digits to a given<br>\ndigit_sum.<br>\n<strong>Note</strong>: permutation are not counted (but you can compute them using factorial)</p>\n<table>\n<thead>\n<tr>\n<th><em>digit_sum</em></th>\n<th><em>x3</em></th>\n<th><em>x4</em></th>\n<th><em>x5</em></th>\n<th><em>row_sum</em></th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td>0</td>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td><strong>3</strong></td>\n</tr>\n<tr>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td>1</td>\n<td><strong>3</strong></td>\n</tr>\n<tr>\n<td>2</td>\n<td>2</td>\n<td>2</td>\n<td>2</td>\n<td><strong>6</strong></td>\n</tr>\n<tr>\n<td>3</td>\n<td>3</td>\n<td>3</td>\n<td>3</td>\n<td><strong>9</strong></td>\n</tr>\n<tr>\n<td>4</td>\n<td>4</td>\n<td>5</td>\n<td>5</td>\n<td><strong>14</strong></td>\n</tr>\n<tr>\n<td>5</td>\n<td>5</td>\n<td>6</td>\n<td>7</td>\n<td><strong>18</strong></td>\n</tr>\n<tr>\n<td>6</td>\n<td>7</td>\n<td>9</td>\n<td>10</td>\n<td><strong>26</strong></td>\n</tr>\n<tr>\n<td>7</td>\n<td>8</td>\n<td>11</td>\n<td>13</td>\n<td><strong>32</strong></td>\n</tr>\n<tr>\n<td>8</td>\n<td>10</td>\n<td>15</td>\n<td>18</td>\n<td><strong>43</strong></td>\n</tr>\n<tr>\n<td>9</td>\n<td>12</td>\n<td>18</td>\n<td>23</td>\n<td><strong>53</strong></td>\n</tr>\n<tr>\n<td>10</td>\n<td>13</td>\n<td>22</td>\n<td>29</td>\n<td><strong>64</strong></td>\n</tr>\n<tr>\n<td>11</td>\n<td>14</td>\n<td>25</td>\n<td>35</td>\n<td><strong>74</strong></td>\n</tr>\n<tr>\n<td>12</td>\n<td>15</td>\n<td>30</td>\n<td>43</td>\n<td><strong>88</strong></td>\n</tr>\n<tr>\n<td>13</td>\n<td>15</td>\n<td>32</td>\n<td>50</td>\n<td><strong>97</strong></td>\n</tr>\n<tr>\n<td>14</td>\n<td>15</td>\n<td>36</td>\n<td>58</td>\n<td><strong>109</strong></td>\n</tr>\n<tr>\n<td>15</td>\n<td>15</td>\n<td>38</td>\n<td>66</td>\n<td><strong>119</strong></td>\n</tr>\n<tr>\n<td>16</td>\n<td>14</td>\n<td>41</td>\n<td>74</td>\n<td><strong>129</strong></td>\n</tr>\n<tr>\n<td>17</td>\n<td>13</td>\n<td>41</td>\n<td>81</td>\n<td><strong>135</strong></td>\n</tr>\n<tr>\n<td>18</td>\n<td>12</td>\n<td>43</td>\n<td>88</td>\n<td><strong>143</strong></td>\n</tr>\n<tr>\n<td>19</td>\n<td>10</td>\n<td>41</td>\n<td>93</td>\n<td><strong>144</strong></td>\n</tr>\n<tr>\n<td>20</td>\n<td>8</td>\n<td>41</td>\n<td>98</td>\n<td><strong>147</strong></td>\n</tr>\n<tr>\n<td>21</td>\n<td>7</td>\n<td>38</td>\n<td>101</td>\n<td><strong>146</strong></td>\n</tr>\n<tr>\n<td>22</td>\n<td>5</td>\n<td>36</td>\n<td>102</td>\n<td><strong>143</strong></td>\n</tr>\n<tr>\n<td>23</td>\n<td>4</td>\n<td>32</td>\n<td>102</td>\n<td><strong>138</strong></td>\n</tr>\n<tr>\n<td>24</td>\n<td>3</td>\n<td>30</td>\n<td>101</td>\n<td><strong>134</strong></td>\n</tr>\n<tr>\n<td>25</td>\n<td>2</td>\n<td>25</td>\n<td>98</td>\n<td><strong>125</strong></td>\n</tr>\n<tr>\n<td>26</td>\n<td>1</td>\n<td>22</td>\n<td>93</td>\n<td><strong>116</strong></td>\n</tr>\n<tr>\n<td>27</td>\n<td>1</td>\n<td>18</td>\n<td>88</td>\n<td><strong>107</strong></td>\n</tr>\n<tr>\n<td><em>col_sum</em></td>\n<td><strong>220</strong></td>\n<td><strong>662</strong></td>\n<td><strong>1483</strong></td>\n<td><strong>2365</strong></td>\n</tr>\n</tbody>\n</table>\n<p>ie, for digit_sum=7 and 4 digits (a+b+c+d=7), there are 11 possibilites.</p>\n<p>Here the json file : <a href=\"https://pastebin.com/TTwrSu6f\" target=\"_blank\">https://pastebin.com/TTwrSu6f</a></p>\n<pre><code>&gt;&gt;&gt; json_data[4][7] \n[(1, 1, 2, 3),\n (0, 1, 1, 5),\n (0, 0, 0, 7),\n (0, 1, 3, 3),\n (1, 2, 2, 2),\n (0, 0, 2, 5),\n (1, 1, 1, 4),\n (0, 0, 1, 6),\n (0, 2, 2, 3),\n (0, 0, 3, 4),\n (0, 1, 2, 4)]\n</code></pre>",
      "rawMarkdown": "Here are a table that give how many way to add up 3, 4 and 5 digits to a given\ndigit_sum.\n**Note**: permutation are not counted (but you can compute them using factorial)\n\n| *digit_sum*        |   *x3* |   *x4* |   *x5* |   *row_sum* |\n|:--------|-----:|-----:|-----:|----------:|\n| 0       |    1 |    1 |    1 |         **3** |\n| 1       |    1 |    1 |    1 |         **3** |\n| 2       |    2 |    2 |    2 |         **6** |\n| 3       |    3 |    3 |    3 |         **9** |\n| 4       |    4 |    5 |    5 |        **14** |\n| 5       |    5 |    6 |    7 |        **18** |\n| 6       |    7 |    9 |   10 |        **26** |\n| 7       |    8 |   11 |   13 |        **32** |\n| 8       |   10 |   15 |   18 |        **43** |\n| 9       |   12 |   18 |   23 |        **53** |\n| 10      |   13 |   22 |   29 |        **64** |\n| 11      |   14 |   25 |   35 |        **74** |\n| 12      |   15 |   30 |   43 |        **88** |\n| 13      |   15 |   32 |   50 |        **97** |\n| 14      |   15 |   36 |   58 |       **109** |\n| 15      |   15 |   38 |   66 |       **119** |\n| 16      |   14 |   41 |   74 |       **129** |\n| 17      |   13 |   41 |   81 |       **135** |\n| 18      |   12 |   43 |   88 |       **143** |\n| 19      |   10 |   41 |   93 |       **144** |\n| 20      |    8 |   41 |   98 |       **147** |\n| 21      |    7 |   38 |  101 |       **146** |\n| 22      |    5 |   36 |  102 |       **143** |\n| 23      |    4 |   32 |  102 |       **138** |\n| 24      |    3 |   30 |  101 |       **134** |\n| 25      |    2 |   25 |   98 |       **125** |\n| 26      |    1 |   22 |   93 |       **116** |\n| 27      |    1 |   18 |   88 |       **107** |\n| *col_sum* |  **220** |  **662** | **1483** |      **2365** |\n\nie, for digit_sum=7 and 4 digits (a+b+c+d=7), there are 11 possibilites.\n\nHere the json file : https://pastebin.com/TTwrSu6f\n```\n>>> json_data[4][7] \n[(1, 1, 2, 3),\n (0, 1, 1, 5),\n (0, 0, 0, 7),\n (0, 1, 3, 3),\n (1, 2, 2, 2),\n (0, 0, 2, 5),\n (1, 1, 1, 4),\n (0, 0, 1, 6),\n (0, 2, 2, 3),\n (0, 0, 3, 4),\n (0, 1, 2, 4)]\n```",
      "votes": null
    },
    {
      "id": "1722243",
      "postDate": "03/14/2022 10:55:12",
      "content": "<p>Awesomeness </p>",
      "rawMarkdown": "Awesomeness",
      "votes": null
    }
  ],
  "comments": [
    {
      "id": 1722243,
      "author_name": "dschettler8845",
      "author_url": "",
      "post_date": "03/14/2022 10:55:12",
      "content": "<p>Awesomeness </p>",
      "votes": null,
      "replies": []
    }
  ],
  "raw_markdown_by_id": {
    "1721364": "Here are a table that give how many way to add up 3, 4 and 5 digits to a given\ndigit_sum.\n**Note**: permutation are not counted (but you can compute them using factorial)\n\n| *digit_sum*        |   *x3* |   *x4* |   *x5* |   *row_sum* |\n|:--------|-----:|-----:|-----:|----------:|\n| 0       |    1 |    1 |    1 |         **3** |\n| 1       |    1 |    1 |    1 |         **3** |\n| 2       |    2 |    2 |    2 |         **6** |\n| 3       |    3 |    3 |    3 |         **9** |\n| 4       |    4 |    5 |    5 |        **14** |\n| 5       |    5 |    6 |    7 |        **18** |\n| 6       |    7 |    9 |   10 |        **26** |\n| 7       |    8 |   11 |   13 |        **32** |\n| 8       |   10 |   15 |   18 |        **43** |\n| 9       |   12 |   18 |   23 |        **53** |\n| 10      |   13 |   22 |   29 |        **64** |\n| 11      |   14 |   25 |   35 |        **74** |\n| 12      |   15 |   30 |   43 |        **88** |\n| 13      |   15 |   32 |   50 |        **97** |\n| 14      |   15 |   36 |   58 |       **109** |\n| 15      |   15 |   38 |   66 |       **119** |\n| 16      |   14 |   41 |   74 |       **129** |\n| 17      |   13 |   41 |   81 |       **135** |\n| 18      |   12 |   43 |   88 |       **143** |\n| 19      |   10 |   41 |   93 |       **144** |\n| 20      |    8 |   41 |   98 |       **147** |\n| 21      |    7 |   38 |  101 |       **146** |\n| 22      |    5 |   36 |  102 |       **143** |\n| 23      |    4 |   32 |  102 |       **138** |\n| 24      |    3 |   30 |  101 |       **134** |\n| 25      |    2 |   25 |   98 |       **125** |\n| 26      |    1 |   22 |   93 |       **116** |\n| 27      |    1 |   18 |   88 |       **107** |\n| *col_sum* |  **220** |  **662** | **1483** |      **2365** |\n\nie, for digit_sum=7 and 4 digits (a+b+c+d=7), there are 11 possibilites.\n\nHere the json file : https://pastebin.com/TTwrSu6f\n```\n>>> json_data[4][7] \n[(1, 1, 2, 3),\n (0, 1, 1, 5),\n (0, 0, 0, 7),\n (0, 1, 3, 3),\n (1, 2, 2, 2),\n (0, 0, 2, 5),\n (1, 1, 1, 4),\n (0, 0, 1, 6),\n (0, 2, 2, 3),\n (0, 0, 3, 4),\n (0, 1, 2, 4)]\n```",
    "1722243": "Awesomeness"
  },
  "source": "meta"
}