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A Zinger Packaging Box Puzzle: How Many Possible Combinations?

A Zinger Packaging Box Puzzle: How Many Possible Combinations?

A customer ordered fifteen Zingers that are to be placed in packages of four, three or one. In how many ways can this order be fulfilled?

Solution to Zinger packaging box puzzle

What is known:

  • More than one box of a specific size to be used in every combination.
  • Any number of specific size package may be used in a combination.
  • No restriction on not using any specific size box in a combination.

The puzzle boils down to finding all possible ways a sum of 15 can be achieved by using additions of numbers of 4, 3 and 1.

A systematic method to find all possible combinations of packaging

Strategy:

  • Find all possible combinations using 4.
  • Find all possible combinations using 3 and 1 excluding 4 (already found).
  • Find all possible combinations using 1, excluding 4 and 3 (already found).

Packages using 4 Zinger box: in each product, first term is the number of boxes and second term the size of box:

  • 3x4 + 1x3 = 15: maximum possible three 4 Zinger boxes with maximum one 3 Zinger box,
  • 3x4 + 3x1 = 15: three four Zinger boxes, zero 3 Zinger box and three one Zinger box,
  • 2x4 + 2x3 + 1x1 = 15, number of 4 Zinger boxes reduced by one, maximum two 3 Zinger boxes used ,
  • 2x4 + 1x3 + 4x1 = 15, with same two 4 Zinger boxes, number of 3 Zinger boxes reduced by 1,
  • 2x4 + 7x1 = 15, zero 3 Zinger box used,
  • 1x4 + 3x3 + 2x1 = 15, number of 4 Zinger boxes reduced further by 1 to the minimum of 1,
  • 1x4 + 2x3 + 5x1 = 15,
  • 1x4 + 1x3 + 8x1 = 15,
  • 1x4 + 11x1 = 15, no 3 Zinger box used.

These are all possible combinations using 4 Zinger box. Total number 9.

Packages using 3 Zinger box and 1 Zinger box:

  • 5x3 = 15, maximum five 3 Zinger boxes used,
  • 4x3 + 3x1= 15, number of 3 Zinger boxes reduced by 1, same with next combinations,
  • 3x3 + 6x1 = 15,
  • 2x3 + 9x1 = 15,
  • 1x3 + 12x1 = 15.

Total 5. Cumulative 14.

Packages using only 1 Zinger box:

  • 15x1 = 15, maximum fifteen 1 Zinger boxes needed.

Total 1.

Cumulative 15 possible package combinations. These are all possible.


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