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#45 Square differences

Set by Alison Kiddle

The number 9 can be written as a difference of two squares as 3² – 0² = 9 – 0 = 9. Can you find a different way to express it as the difference of two squares?

Which numbers between 1 and 30 can be written as a difference of two squares?

Some numbers cannot be written in this way – can you determine which ones?

Solution next week

#44 Dice and cards

Solution

The average for three dice is 10.5. The sum of the values on one die is 21, which is divided by 6 to give the average value for a single die as 3.5. The dice are independent, so the average value of the sum of the three is 3.5 × 3 = 10.5. For cards drawn with replacement, the answer is also 10.5 by following the same logic. If we draw three cards without replacement, there are 20 combinations (we don’t care about their order, as it doesn’t matter for a sum). Taking the sum of each, which ranges from 6 to 15, and finding the average, also comes out as 10.5. For an easier approach, we can note that across all 20 sums the same six numbers (1 to 6) are repeated 10 times. So the average is simply the sum of 1 to 6 multiplied by 10, then divided by 20 to give 10.5.