Q:

The clock with a light fixture.

This is a problem from the test given out by my teacher which I didn't manage to solve any of the problems. This one, in particular, is the problem I decided to turn into a story. I apologized in advance if there are any minor inconveniences.

There was once a kid who has a magnificent clock which is his "best friend", he carried it everywhere with him.
Although the clock doesn't have any numbers on its face, it is accompanied by a light bulb on every spoke indicated where the numbers were supposed to be.
This particular clock doesn't function like every other clock, its only purpose was to be a decoration and make the boy happy.
Perhaps there is one interesting mechanics inside the clockwork. Every time the boy put the hands of the clock into a position such that between the hands were exactly 3 light bulbs, the lights would turn if it is off and vice versa.
But in the gamily he lived in there was his disobedient elder step-brother. He, who was obsessed with the applications of logic in technology, would always find ways to annoy his siblings.
Knowing that his brother has the traits of a perfectionist. He started hiding socks from him, opening the closet slightly in his bedroom at night and adding a lot of salt into his cups of milk at breakfast.
One time, when the boy forgot to bring his "best friend" to the local market with his mother, the step-brother went into his bedroom, broke open the inner mechanics and only turn on the first light where the number 1 should have been and put the clock back together before anyone could notice.
When the boy came home, he ran up the stairs to check on the clock and realised that his brother had destroyed his clock. He begged for his sibling to fix the clock.
"Under one condition should I repair the clock for you. If you could make all the lights turned off except the second light then I will fix it. And no, you can not just turn the clock 30 degrees clockwise and say that you're done." - the elder brother said.
The boy tried every possible way to make it happen but he couldn't. He cried in his room all afternoon. His mother promised him to buy another one but he refused, saying that it's not worth her money.
Is there a way to turn off the first light and turn on the second light without beaking open the inner mechanics?

The answer is obviously no. But I don't know how to prove it.

A:

In simpler terms, the state of the clock is an element of $V:=\Bbb F_2^{12}$, and by manipulating the clock hands, we can add any element of the form $v_i:=e_i+e_{i+1}+e_{i+2}$, $1\le i\le 12$, where $e_1,\ldots, e_{12}$ form the canonical basis and $e_{13}=e_1$, $e_{14}=e_2$ is understood.
Let $W\subseteq V$ be the subspace spanned by the $v_i$. Then the problem statement asks if $e_1+e_2\in W$.
In this form, the problem is readily solved: Each $v_i$ is orthogonal to $$e_1+e_3+e_4+e_6+e_7+e_9+e_{10}+e_{12},$$ hence so are all $w\in W$, but $e_1+e_2$ is not.

