Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?
a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)
During the election for the government of the planet of Liars and
Truth-Tellers, \(12\) candidates each
gave a short speech about themselves.
After everyone had spoken, one alien said: “So far, only one lie has
been told today.”
Then another said: “And now two have been said so far.”
The third said: “And now three lies have been told so far,” and so on —
until the twelfth alien said: “And now twelve lies have been told so
far.”
It turned out that at least one candidate had correctly counted how many
lies had been told before their own statement.
How many lies were said that day in total?
Long before meeting Snow White, the seven dwarves lived in seven different mines. There is an underground tunnel connecting any two mines. All tunnels were separate, so you could not start in one tunnel and somehow end up in another. Is it possible to walk through every tunnel exactly once without retracing your path?
There is a queue of \(n\) truth tellers and liars. The first person says, “more than half of us are liars". The second person says, “more than a quarter of us are liars". The third person says, “more than an eighth of us are liars", and so on, until the \(n\)th person says, “more than \(\frac{1}{2^n}\) of us are liars". Describe what the number of truth-tellers and liars could be, as well as their placement in the queue. Note that the solutions are not fixed numbers.