Daily Math Puzzle: 2026-09-12
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2026-09-12
You've finally acquired the treasure map to the legendary 'Crimson Ruby'! It instructs you to start at the ancient well (0,0) and reach the hidden X-mark at coordinates (3,3) by taking a series of steps. The map explicitly states you must take exactly 3 steps North and exactly 3 steps East. However, a cryptic inscription nearby warns: 'No two consecutive steps may be in the same cardinal direction.' (For example, if you step North, your very next step cannot also be North). How many *unique valid paths* can you take from the well to the X-mark, strictly following these instructions?
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Solution
2 — To reach coordinates (3,3) from (0,0) using only North (N) and East (E) steps, you must take exactly 3 steps North and 3 steps East, for a total of 6 steps.
The crucial rule states: 'No two consecutive steps may be in the same cardinal direction.' This means you cannot take N-N or E-E. Every step must alternate direction.
Let's construct the possible valid paths:
1. **Starting with a North (N) step:**
* Since you cannot take N-N, the second step *must* be East (E).
* Since you cannot take E-E, the third step *must* be North (N).
* This pattern continues: N E N E N E.
This path uses exactly 3 N and 3 E steps, and perfectly alternates directions. This is one valid path.
2. **Starting with an East (E) step:**
* Since you cannot take E-E, the second step *must* be North (N).
* Since you cannot take N-N, the third step *must* be East (E).
* This pattern continues: E N E N E N.
This path also uses exactly 3 N and 3 E steps, and perfectly alternates directions. This is a second valid path.
No other sequences are possible, because once the first step is chosen (either N or E), the alternating rule strictly dictates every subsequent step to ensure no consecutive steps are in the same direction, and all 3 N and 3 E steps are used.
Therefore, there are only 2 unique valid paths.
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