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September 2017 Puzzle Periodical - Arabella's Spider Web

By Katrina J., NSA Mathematician

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Can you help Arabella find her bugs on the leaves in her Spider Web?

Problem:

Arabella the Spider is saving food for the long winter. Arabella wants to store the bugs she caught on 26 fallen leaves, so she can find them later. But, Arabella doesn't want to waste time by going through any leaves more than once.

In Arabella's original web, Arabella can't get to all of the leaves without crossing some of them more than once. But, if Arabella adds just one web between two of the leaves, she can get to every leaf without repeating. There are four different pairs of leaves that Arabella could connect to solve her problem. Can you find all four possible solutions?

Note: Arabella may take any path she chooses as long as she begins on leaf 1 and ends on leaf 26.

Bonus Puzzle:

Can you show why Arabella cannot get to every leaf without repeats on her web as it is now?

Arabella the Spider has a web connecting 26 numbered leaves. The following list describes the connections.

  • Leaf 1 - connects to Leaves 6 and 14.
  • Leaf 2 - connects to Leaves 8, 11, and 26.
  • Leaf 3 - connects to Leaves 5, 18, and 23.
  • Leaf 4 - connects to Leaves 16, 19, 22, and 24.
  • Leaf 5 - connects to Leaves 3 and 9.
  • Leaf 6 - connects to Leaves 1, 14, and 23.
  • Leaf 7 - connects to Leaves 10, 11, and 23.
  • Leaf 8 - connects to Leaves 2, 11, and 21.
  • Leaf 9 - connects to Leaves 5, 12, and 17.
  • Leaf 10 - connects to Leaves 7, 13, and 22.
  • Leaf 11 - connects to Leaves 2, 7, and 8.
  • Leaf 12 - connects to Leaves 9, 20, and 25.
  • Leaf 13 - connects to Leaves 10 and 16.
  • Leaf 14 - connects to Leaves 1, 6, and 23.
  • Leaf 15 - connects to Leaves 21 and 26.
  • Leaf 16 - connects to Leaves 4, 13, and 19.
  • Leaf 17 - connects to Leaves 9 and 20.
  • Leaf 18 - connects to Leaves 3 and 25.
  • Leaf 19 - connects to Leaves 16 and 24.
  • Leaf 20 - connects to Leaves 12, 17, and 25.
  • Leaf 21 - connects to Leaves 8 and 15.
  • Leaf 22 - connects to Leaves 4 and 10.
  • Leaf 23 - connects to Leaves 3, 6, and 14.
  • Leaf 24 - connects to Leaves 4 and 19.
  • Leaf 25 - connects to Leaves 12, 18, and 20.
  • Leaf 26 - connects to Leaves 2 and 15.

Click to see the answer!

Solution:

This problem has 4 solutions and answers: 5-13, 5-22, 18-13, 18-22. Only the first solution is fully drawn on the solution illustration, but all four are explained below.

Illustration of the solutions with numbered leaves connected by lines representing the spider web. A detailed description of the solution follows in the text presented in the explanation section.

There are minor variations in each path, such as switching the order of leaves 6 and 14, which are unaffected by the solution choice, but additional variations are not listed. The following list describes the connections:

  • Leaves 5 and 13: 1, 6, 14, 23, 3, 18, 25, 12, 20, 17, 9, 5, 13, 16, 19, 24, 4, 22, 10, 7, 11, 2, 8, 21, 15, 26;
  • Leaves 5 and 22: 1, 6, 14, 23, 3, 18, 25, 12, 20, 17, 9, 5, 22, 4, 24, 19, 16, 13, 10, 7, 11, 2, 8, 21, 15, 26;
  • Leaves 18 and 13: 1, 6, 14, 23, 3, 5, 9, 17, 20, 12, 25, 18, 13, 16, 19, 24, 4, 22, 10, 7, 11, 2, 8, 21, 15, 26;
  • Leaves 18 and 22: 1, 6, 14, 23, 3, 5, 9, 17, 20, 12, 25, 18, 22, 4, 24, 19, 16, 13, 10, 7, 11, 2, 8, 21, 15, 26.

Explanation:

First, let's re-draw the situation to be less messy! Once we've organized the leaves a bit, we see that there are 4 distinct groups of leaves which can only be accessed via one leaf: Group 1 is only accessible via leaf 23, Group 2 via leaf 3, Group 3 via leaf 10, and Group 4 via leaf 11.

Here we see the problem - once Arabella has entered a group of leaves, she cannot get out again without crossing the "access leaf" again. For Group 1 this is okay, since Arabella can start on leaf 1, move to leaf 6, then leaf 14, then exit at leaf 23, and doesn't need to go back in. Similarly, Group 4 contains the last leaf, so Arabella can enter the group from leaf 11, then go to leaf 2, leaf 8, leaf 21, leaf 15, and exit at leaf 26.

But for Group 2 and Group 3, Arabella must enter and exit the group. This is where the extra web comes in handy! If we connect a leaf in Group 2 and a leaf in Group 3, then Arabella can go from Group 1 to Group 2 using leaf 23, then Group 2 to Group 3 using the new web, and finally from Group 3 to Group 4 using leaf 7.

However, not just any leaves from Group 2 and Group 3 will work. Arabella must cover all of Group 2 before moving to Group 3, and cover all of Group 3 before going to Group 4. This still allows for a few different combinations: from Group 2 we can use leaf 5 or leaf 18; from Group 3 we can use leaf 13 or leaf 22.

Here are the different paths Arabella could take for each combination (there are minor variations in all of the paths, such as switching the order of leaves 6 and 14, which are unaffected by the solution choice, but we will not list those):

  • Leaves 5 & 13 -- 1, 6, 14, 23, 3, 18, 25, 12, 20, 17, 9, 5, 13, 16, 19, 24, 4, 22, 10, 7, 11, 2, 8, 21, 15, 26
  • Leaves 5 & 22 -- 1, 6, 14, 23, 3, 18, 25, 12, 20, 17, 9, 5, 22, 4, 24, 19, 16, 13, 10, 7, 11, 2, 8, 21, 15, 26