
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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Let  $ n$  and  $ k$  be positive integers with  $ k \geq n$  and  $ k \minus{} n$  an even number. Let  $ 2n$  lamps labelled  $ 1$ ,  $ 2$ , ...,  $ 2n$  be given, each of which can be either *on* or *off*. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let  $ N$  be the number of such sequences consisting of  $ k$  steps and resulting in the state where lamps  $ 1$  through  $ n$  are all on, and lamps  $ n \plus{} 1$  through  $ 2n$  are all off.

Let  $ M$  be number of such sequences consisting of  $ k$  steps, resulting in the state where lamps  $ 1$  through  $ n$  are all on, and lamps  $ n \plus{} 1$  through  $ 2n$  are all off, but where none of the lamps  $ n \plus{} 1$  through  $ 2n$  is ever switched on.

Determine  $ \frac {N}{M}$ .


*Author: Bruno Le Floch and Ilia Smilga, France*

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