Problems

Let \(x,y\) be nonnegative integers. Determine when \(\text{Nim}(x,y)\) is a losing position and when it is a winning position.

Is \(\text{Nim}(1,2,3)\) a winning position or a losing position?

Is \(\text{Nim}(1,2,4,5,5)\) a winning position or a losing position?

Let us define XOR (or addition mod 2). XOR is defined for 0 and 1 only. Here is a table recording the values of XOR:

Now we define the important concept of nim-sum. Given two natural numbers \(x\) and \(y\), we first convert them into binary representations and then compute XOR on individual digits. The resulting number, denoted \(x \oplus y\), is the nim-sum of \(x\) and \(y\). Here is an example.

This is simply saying \(22 \oplus 5 = 19\). Note that \(22=(10110)_2\) and \(5=(00101)_2\).

Verify \((x \oplus y) \oplus z = x \oplus (y \oplus z)\), so we can speak of \(x \oplus y \oplus z\) with no ambiguity.

Show that \(x \oplus y = 0\) if and only if \(x = y\). Remember that \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).

Show that \(\text{Nim}(x,y,z)\) is a losing position if and only if \(x \oplus y \oplus z = 0\). Remember that \(x \oplus y\) denotes the nim-sum of \(x\) and \(y\).

Is \(\text{Nim}(7,11,15)\) a winning position or a losing position? If it is a winning position, what is the optimal move?

Two fractions sum up to \(1\), but their difference is \(\frac1{10}\). What are they?

On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by \(20\)th birthday. From that, take away the year I was \(30\). The result will be \(16\)." How old is my grandma?

In the long addition above, each letter corresponds to a different digit. What is the sum \(D + O +G + C +A +T\)?