[OlympiadGRN20150406] Grid Ratio Ninja

Perhaps you are an expert at Fruit Ninja, slashing fruits mercilessly with precision using your much-feared finger.  Now can you do the mathematical equivalent below?

Question

Solution

     We would be getting nowhere fast if we took a random approach.  We are only allowed two slashes.  Let us look at the question for more clues.  We have a  5  by  6  grid, giving a total of  30  cells.  The total number of parts in the ratio is  1 + 2 + 3 + 4 = 10.  10  parts correspond to  30 cells,  so each part is  3  cells.  So the number of cells in each portion are  3,  6,  9  and  12.

     Note that if we pair up  3  with  12  and  6  with  9,  we have  15 cells each.  We can just slash the grid down the middle to achieve this.

     We now need ratios of  1 : 4  and  2 : 3  for the remaining parts.  We certainly can achieve this if we were allowed two more slashes and if we used horizontal lines, like this:

However, we only have one more slash.  Can we make this slash do the work of the two horizonal slashes?  Yes!  Replace the two horizontal slashes with a single slanting slash that passes through the mid-points of the horizontal slashes.  This gives four trapezia.  The reason why it works is because for each portion, an extra triangular area is added but it is offset by the removal of a triangle of the same area, like this:-

We can check that the ratios of the areas of the portions are as required.  Done!

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H08. Make suppositions

H09. Restate the problem in another way

H11. Solve part of the problem