<p>.........Column 1 Column 2 Total
Row 1 ________ ________ 53</p>
<p>Row 2 ________ ________ 26</p>
<p>Row 3 ________ ________ 21</p>
<p>Total......36............64........100</p>
<p>In the table above, each of the eix empty boxes should contain a number entry so that the column and row totals are as given. Juan wants to complete the table. What is the least number of entries that he must make for in order to complete the table?
A) One
B) Two
C) Three
D) Four
E) Six</p>
<p>It is not clear exactly what you are asking, but perhaps you are asking for the smallest number of entries you must be given in order to complete the table uniquely. It depends whether negative numbers are allowed. If they are allowed, the answer is two. First convince yourself that being given just one entry does not do it. If (RI,C1) = 12, say, we know that (R1,C2) = 41, but all we know beyond that is (R2,C1)+(R3,C1) = 24 and (R2,C2)+(R3,C2) = 23. Now we just need to show that getting two entries works. If you have two entries in one column, you are done. This tells you the final entry in the column and then you can get the entries in each corresponding row. </p>
<p>If negative numbers are not allowed, the smallest number could be one. If you are given (R1,C1) = 36, then you must have (R2,C1) = (R3,C1) = 0, which allows you to fill in all remaining numbers. If (R1,C1) < 36, the answer is again two.</p>