¿ìè¶ÌÊÓƵ

#44 Dice and cards

Set by Christopher Dearlove

On each turn of a game, we roll three standard dice with faces numbered 1 to 6 and add up the numbers shown. What is the average (mean) value of that total over a long game?

Now suppose instead of rolling dice, we draw three cards from a six-card deck where the cards are numbered 1 to 6. If we put each card back after drawing it and shuffle before drawing another card, does this change the expected value?

If we instead draw three cards without replacement (so the probabilities are no longer independent), what is the average value of the sum?

Solution next week

#43 Consecutive sums

Solution

We can write 14 = 2 + 3 + 4 + 5.

Any odd number can be written as the sum of two consecutive numbers: 2n + 1 = n + (n + 1).

Even numbers that have an odd factor like 3, 5, 7 etc, can be written as a sum of that many terms: e.g. 12, which is 4 × 3, can be written as three numbers centred on 4: 3 + 4 + 5. In the case of 14, we use seven numbers centred on 2, but since the first three of these are -1, 0 and 1, they cancel out, leaving us with the sum above.

The only numbers that can’t be expressed this way are those without an odd factor (other than 1) – exactly the powers of 2: 1, 2, 4, 8, 16 and so on.