Two circles and a line riddle: Can you traverse all parts of the puzzle figure without lifting your pencil, revisiting a part, or crossing a line?

**Time to solve:** 15 minutes.

### Solution to the Traversing two circles and a line riddle: can you traverse?

To explain how to solve, the four crossing points between the vertical line and the two concentric circles are labeled as shown.

Starting the journey either from ‘a’ or from ‘b’ will be the same.

Start from point ‘a’ chosen. Only one way to go—upwards to point ‘c’.

Finally, the journey must end at ‘b’ in the upward direction from ‘d’ to ‘b’. But for now, what should be the next move from ‘c’?

Time to analyze the **strategy of traversing** the two circles.

Critical question:Can we traverse a full circle at one go, say starting from ‘c’ moving along the left half of the larger circle in upward direction to ‘d’, crossing over to the right half of the circle and moving downward to complete the larger circle traverse at ‘c’ again?

What will happen if we do this?

If we do this, we won’t be able to cross the continuous line of the periphery of the circle at point ‘d’ when we try to end the journey going upwards to point ‘b’.

Conclusion 1:The larger circlecannot be traversed continuouslyat one go.

Then how to traverse the circle?

**The breakthrough idea** is,

Conclusion 2 Strategy:The larger circle must be traversed half-circle by half-circle. After covering a half-circle,direction must be changed to vertical(upward or downward). The other half of the circle will be left to cover later.

Conclusion 3:As the vertical directions will be used for changing direction AFTER traversing a half-circle, this direction cannot be used from ‘c’ at the start. Next move must be along either the left or the right half of the larger circle.

Choose to move along the left half of the larger circle upwards to point ‘d’.

And as per strategy, move next VERTICALLY DOWN to point ‘f’ without crossing over to the right half of the circle.

Result:The right half of the larger circle is left to cover later.

How to traverse it? Best way is to start from ‘c’ and end at ‘d’. Next, turn upwards and complete the journey at ‘b’.

An important question arises,

Critical question:Looking forward to the end state,can we decide from which point to arrive at point ‘c’ and then move on to the right half of the large circle?

Answer:It must be from point ‘e’ downwards to point ‘c’.

So in the last part of the journey, we must reach the point ‘e’ **after completing the inner circle and the vertical line segment inside the inner circle.**

Where are we at present? We stopped at point ‘f’ after covering the left half of the larger circle.

**Question:** Can we now go straight down to ‘e’, turn left or right to complete the full inner circle at one go, and come back to point ‘e’ again?

No, we cannot because the circular path will cross the continuous vertical line from ‘d’ to ‘e’.

Conclusion 4:We must turn left (or right) along the left (or right) half of the inner circle and move down to point ‘e’. No other alternative.

Let us choose to move along the left half of the inner circle through point ‘j’ to point ‘e’.

**Question:** Can we now cross over to the right half of inner circle to complete the traversal of the inner circle at one go?

Again, it is not possible. Because, on the next part of the journey from point ‘f’ to point ‘c’ along the vertical direction, the path will cross the continuous periphery of the inner circle at point ‘e’.

Conclusion 5:The only way to move next will be upwards along the vertical line segment to point ‘f’. Turn right and down along the right half of the inner circle and reach point ‘e’. Turn again downwards to point ‘c’ and then complete the journey along right half of the larger circle to point ‘d’ and then on to point ‘b’.

The solution path is shown.

The full path for successful traversal of all the parts of the graphic is,

a → c → g → d → f → j → e → f → k → e → c → h → d → b.

#### End note

The key breakthrough idea is—*breaking up each circle into two half-circles, as well as breaking up the vertical line into segments at each crossing point.*

This is achieved by analyzing the consequences of taking action at junction points. It is **consequence analysis.** It helps easy solution to even many real-life problems.

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