Magic Caterpillars

455 /www/nrich/html/content/03/02/15plus1/ 1 2011-02-01T00:00:01 <mdoxml version="1.0">In the Land of Trees all the caterpillars have numbers on their feet and hips (vertices) and on their legs and body segments (edges) as shown on this 4 legged caterpillar. All the whole numbers from 1 to $v+e$ are used where $v$ is the number of vertices and $e$ is the number of edges. Biologists classify them by their vertex-sums. A vertex sum is the total of the numbers on the vertex and all the edges at that vertex. The caterpillar shown has vertex sums: 11 (8+3), 13 (9+4), 15 (10+5), 17 (11+6), 25 (8+9+7+1) and 30 (7+10+11+2). <mdo:image alt="Magic Caterpillar" height="194" src="magic_c.gif" width="370"></mdo:image> Show that one day a biologist may find a rare magic 4-legged caterpillar having the same sum at all its vertices and describe this creature. Could there be two species of magic 4-legged caterpillars with different numberings? Do magic 6-legged caterpillars exist? What about magic caterpillars with even more legs? </mdoxml> <mdoxml version="1.0">Peter sent us this solution, using the starting point we gave: There are $6$ vertices and $5$ edges. Let's suppose the magic constant is $S$ and this is the same at each vertex. The total of all the numbers $1 + 2 + ... + 11 = 66$ but the numbers on the edges, $a,b,c,d,e$ are counted twice. So, adding the magic sum at all six vertices: $$(a+b+c+d+e)+66=6S$$ So $a+b+c+d+e$ is a multiple of $6$ and it is at least $1+2+3+4+5=15$ so it must be $18$ or more. Hence $6S\geq84$ and $S\geq14$. We're going to call the horizontal edge $e$, the front two edges $a$ and $b$, and the back two legs $c$ and $d$. Also, we see that the numbers $11$, $10$ and $9$ must all be in feet, as they're too big to be anywhere else. Let's try $S=14$ first. We may as well take $a=3$, as the legs are all interchangeable at this stage. Now if $b=4$, then we have $c=5$ (as the back legs are interchangeable). Also, we must have $d$ and the back foot as $6$ and $8$ (in some order), and we soon see that this isn't possible (as we'd also need $e$ plus the top left hip to add up to $7$). If $b$ isn't $4$, then we can assume that $c=4$. But similarly we find that we can't have $b=9$ or $b=8$ or $b=7$. So $S$ isn't $14$. Let's try $S=16$. But then we need four pairs of numbers that sum to $16$ (one for each leg). We try: $(11,5)$, $(10,6)$, $(9,7)$, $(8,?)$ and we can't repeat the $8$ so that isn't possible. So let's try $S=15$. This time, we may as well take $a=4$. There are only $4$ pairs that add up to $15$: $(11,4)$, $(10,5)$, $(9,6)$ and $(8,7)$. Call the number in the top left hip $f$. If $b=5$, then we have $c=6$ and $e+f=6$, which isn't possible (as we don't have the right numbers left). So we can take $c=5$. If $b=8$, have $e+f=3$ and then we'll need $f+g=4$, and this isn't possible. If $b=7$, have $e+f=4$ and we also have $f+g=4$ and this isn't possible either. Also we can't have $d=8$, as this is too big. So the only possibility is the one shown below (and this does indeed work). (When I say the only possibility, of course the front legs could be switched, or the back legs, or the front and back swapped, but it's really the same caterpillar.) <div><mdo:image alt="Magic caterpillar" height="194" src="magic_c_solution.gif" width="370"></mdo:image></div> <div> </div> <div>West Lakes Academy year 10 MadMathsClub tried to prove our claim that there was no magic 6-legged caterpillar. In fact, we were wrong! They found a magic 6-legged creature as shown below:</div> <div> </div> <div><mdo:image alt="" src="order6magiccaterpillar.GIF" style="width: 555px; height: 268px;"></mdo:image></div> <div>whose vertex-sums are all 23. Good work!</div></mdoxml> <mdoxml version="1.0"> The numbers on the edges are counted twice. The reasoning needed here is much the same as for the problem Plum Tree. See the <a href="http://nrich.maths.org/public/viewer.php?obj_id=456&part=clue"> detailed hint for Plum Tree.</a><a href="http://nrich.maths.org/public/viewer.php?obj_id=455&part=clue"></a> </mdoxml> 2 5 0 0 0 1 1 Magic Caterpillars Put numbers 1 to n on the edges and vertices of a graph so that the sum of the numbers on a vertex and on all arcs joined to that vertex is the same for all vertices. Algebra Creating expressions/formulae Decision Mathematics and Combinatorics Networks/Graph Theory Decision Mathematics and Combinatorics Combinatorics Using, Applying and Reasoning about Mathematics Working systematically