Question 1150802
<font face="times" color="black" size="3">
Answer: <font color=red size=4>172</font>


============================================================================

How I got that answer:


List out numbers that have at least one '3' in them and count how many '3's are used up so far.


Below is a handy table to show what I mean. 
Note that the second column steadily increases by 1 each time except in the cases of 33 and 133 which have two copies of '3' (as opposed to one copy of '3').

<table border = "1" cellpadding = "5">
<tr><td>Value</td><td>Number of '3's used so far</td></tr>
<tr><td>3</td><td>1</td></tr>
<tr><td>13</td><td>2</td></tr>
<tr><td>23</td><td>3</td></tr>
<tr><td>30</td><td>4</td></tr>
<tr><td>31</td><td>5</td></tr>
<tr><td>32</td><td>6</td></tr>
<tr><td>33</td><td>8 (note the jump by +2)</td></tr>
<tr><td>34</td><td>9</td></tr>
<tr><td>35</td><td>10</td></tr>
<tr><td>36</td><td>11</td></tr>
<tr><td>37</td><td>12</td></tr>
<tr><td>38</td><td>13</td></tr>
<tr><td>39</td><td>14</td></tr>
<tr><td>43</td><td>15</td></tr>
<tr><td>53</td><td>16</td></tr>
<tr><td>63</td><td>17</td></tr>
<tr><td>73</td><td>18</td></tr>
<tr><td>83</td><td>19</td></tr>
<tr><td>93</td><td>20</td></tr>
<tr><td>103</td><td>21</td></tr>
<tr><td>113</td><td>22</td></tr>
<tr><td>123</td><td>23</td></tr>
<tr><td>130</td><td>24</td></tr>
<tr><td>131</td><td>25</td></tr>
<tr><td>132</td><td>26</td></tr>
<tr><td>133</td><td>28 (note the jump by +2)</td></tr>
<tr><td>134</td><td>29</td></tr>
<tr><td>135</td><td>30</td></tr>
<tr><td>136</td><td>31</td></tr>
<tr><td>137</td><td>32</td></tr>
<tr><td>138</td><td>33</td></tr>
<tr><td>139</td><td>34</td></tr>
<tr><td>143</td><td>35</td></tr>
<tr><td>153</td><td>36</td></tr>
<tr><td>163</td><td>37</td></tr>
<tr><td>173</td><td>38</td></tr>
</table>

The second column stops at 38 and not 37. This will help us determine the ceiling, so to speak, that Patty can reach. 


If she had 38 copies of '3' available, then Patty could form 173. But she doesn't have 38 copies, and instead has 37, so the highest number she can reach is <font color=red>172</font>.


edit: the tutor @math_helper has a much more elegant and simple solution, so it's best to go with that. 
</font>