An irrational number in $(0,1)$ is funny if its first four decimal digits are the same. For example, $0.1111 + e/10^5$ is funny. Prove that $0.1111$ is not a sum of 1111 funny numbers and that every $x\in (0,1)$ is the sum of 1112 different funny numbers.
For the first part, the only irrational numbers that could possibly sum to $0.1111$ must start with $0.0000$ as otherwise the sum would be too large. Then each such number is strictly less than $0.0001$, so the sum of $1111$ such numbers would be less than $0.1111$. Now let $x\in (0,1)$. To find the sum for $x$, first let $k$ be the largest number of the form $0.uuuu$ that is less than $x$ and increase k only by a bit (I'm not sure how much) so that it becomes funny. Then consider $(x-k)/1111$. Now the idea is to slightly adjust these numbers to make them all different and irrational as well as funny, but I'm not sure about the details. k and the modified numbers should produce the desired sum.
I misunderstood the question initially, but it really does mean what it says $-$ the first four digits after the decimal point must all be the same. So all irrational numbers in $(0,0.0001)$ are funny, because they start with four zeroes. And any funny number $>0.1$ must also be $>0.1111$.
So now all the OP needs is a way of finding $1112$ distinct irrational numbers in $(0,0.0001)$ whose sum is $0.1111$. And we can do this as follows:
Let $q$ be the (rational) number $0.1111/1112$. Note that $q<0.0001$. Let $\xi$ be a positive irrational number, and consider the set $$\{q+\xi,q-\xi,q+2\xi,q-2\xi,\ldots,q+556\xi,q-556\xi\}$$ The sum of these $1112$ numbers is $1112q=0.1111$. And if we make $\xi$ small enough, each number will be an irrational number in $(0,0.0001)$, and therefore funny. Specifically, it is enough to choose $\xi$ so that the largest of these numbers, $q+556\xi$, is less than $0.1$. For instance, $$\xi=\frac{0.0001-q}{556\pi}$$