Problems

Michael thinks of a number no less than $1$ and no greater than $1000$. Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking $10$ questions?

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication $1995\times \ast \ast \ast$, which is the number in the fifth row. This is an example of a “long multiplication table”.

Two people play a game with the following rules: one of them guesses a set of integers $(x_1,x_2,\dots ,x_n)$ which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum $a_1{x}_1+\cdots +a_n{x}_n$, where $(a_1,\dots ,a_n)$ is any set. What is the smallest number of questions for which the guesser recognizes the intended set?

Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?

Prove that: $$a_1{a}_2{a}_3\cdots a_{n-1}{a}_n\times {10}^3\equiv a_{n-1}{a}_n\times {10}^3(\mathrm{mod}4),$$ where $n$ is a natural number and $a_i$ for $i=1,2,\dots ,n$ are the digits of some number.

The digits of a 3 digit number $A$ were written in reverse order and this is the number $B$. Is it possible to find a value of $A$ such that the sum of $A$ and $B$ has only odd numbers as its digits?

Let $x$ be a 2 digit number. Let $A$, $B$ be the first (tens) and second (units) digits of $x$, respectively. Suppose $A$ is twice as large as $B$. If we add the square of $A$ to $x$ then we get the square of a certain whole number. Find the value of $x$.

How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.

We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?

Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?