Pathway Math Problem (MPfG 2019 #10)

Math Prize for Girls 2019 #10

Problem: A 1*5 rectangle is split into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second it jumps from its current cell to one of the adjacent cells. The frog makes exactly 15 jumps. How many paths can the frog take to finish at cell 5?

Solution: When we see such a problem, we automatically think of using some counting of permutations to help solve this, but actually, all we need is addition.


      1
      2
      3
      4
     5
Step 1
      1



Step 2      1
      1

Step 3
      2
      1
Step 4      2
      3
      1
Step 5
      5
      4
Step 6      5
      9
      4
Step 7
     14
     13
Step 8     14
     27
     13
Step 9
     41
     40
Step 10     41
     81
     40
Step 11
    122
    121
Step 12    122
    243
    121
Step 13
    365
    364
Step 14    365
    729
    364

On the first step, there is one part to go to the cell 2, and on each later step, we sum the two previous values from the two adjacent cells. This covers all the paths as each step can either to go: the cell to the left, or the cell to the right.

Interesting enough, the middle column (cell 3) has powers of 3 every 2 steps.

When testing a different amount of cells, we see that only in this case are there powers of any kind.

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