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Paths on a Grid (illustration)
FILE No.156Unsolved

Paths on a Grid

Difficulty ★★☆about 5 minLogic PuzzleCombinationsGrids
In a 4×4 grid, from the lower left to the upper right, moving only right or up. How many paths are there?

Case Record

Grid streets are laid out in a 4 × 4 block pattern. From the lower left to the upper right, you move only right or up.

Your question

How many shortest routes are there?

Case Files

Diagram

A 4 × 4 grid of streetsBottom left: startTop right: goalMove only right or up. How many shortest routes are there?
Diagram

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

How many times do you move right, and how many up?

Open hint 2

The order of the 8 moves is the route.

Open hint 3

It is the number of ways to choose 4 from 8.

Solve It

Your question

How many shortest routes are there?

Choose the answer closest to the truth.

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The Truth

70

The answer is "70."

You make 4 moves right and 4 moves up, 8 moves in all. Choose which 4 of the 8 moves are "right." Choosing 4 from 8 gives 70 ways.

16 and 256 come from simply multiplying the choices at each step, and do not use the condition "four of each."

Ways to arrange 4 rights and 4 upsItemDetailMoves4 right + 4 up = 8 movesChooseWhich 4 of the 8 are "right"Count8 × 7 × 6 × 5 ÷ (4 × 3 × 2 × 1)Answer70 routes
Diagram of the solution

Key Points

  • Rephrase the route as a sequence of rights and ups
  • Count by combinations
  • Trying every case is hard without a rule

Why people get fooled

Because you try to trace and count the routes on the map one by one.

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