7160 /www/nrich/html/content/id/7160/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Shakil wants to remove numbers from the set $\{1,2,3,..., 16\}$ so that no two remaining numbers add to make a perfect square. What is the smallest number of numbers that he needs to remove?<br></br> <br></br>  <br></br> If you liked this problem, <a href="http://nrich.maths.org/790">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br> <br></br>   </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> The following seven pairs add to $16$ so at least one of each pair must be removed:<br></br> $(1,15)$, $(2,14)$, $(3,13)$, $(4,12)$, $(5,11)$, $(6,10)$, $(7,9)$.<br></br>  <br></br> If removing these seven is sufficient, then we would be left with $8$, $16$ and seven others.<br></br>  <br></br> But<br></br> <div style="margin-left: 40px;">  $16+9 = 25$      So we must remove $9$    and keep its partner $7$</div> <div style="margin-left: 40px;">    $7+2 = 9$        So we must remove $2$     and keep $14$</div> <div style="margin-left: 40px;">$14+11 = 25$      So we must remove $11$   and keep $5$</div> <div style="margin-left: 40px;">    $5+4 = 9$        So we must remove $4$     and keep $12$</div> <div style="margin-left: 40px;">$12+13 = 25$      So we must remove $13$   and keep $3$</div> <div style="margin-left: 40px;">    $3+1 = 4$        So we must remove $1$     and keep $15$</div> <div style="margin-left: 40px;">$15+10 = 25$      So we must remove $10$   and keep $6$</div> <div> </div> <div>But we have kept $3$ and $6$ which add to $9$.</div> <div> </div> <div>Hence it is not sufficient to remove only seven. If we remove the number $6$, we obtain a set which satisfies the condition: $\{8,16,7,14,5,12,3,15\}$ or in ascending order $\{3,5,7,8,12,14,15,16\}$.</div> <div> </div> <div>Hence eight is the smallest number of numbers that may be removed.</div> <br></br>  <br></br>  <br></br></mdoxml> 2 3 0 0 1 1 0 Weekly Problem 12 - 2011 Numbers and the Number System Square numbers Using, Applying and Reasoning about Mathematics Working systematically Calculations and Numerical Methods Addition & subtraction