本周问题：平方差
数字9可以写成两个平方的差，例如 。你能找到另一种将其表达为两个平方的差的方法吗？
1到30之间的哪些数字可以写成两个平方的差？
有些数字无法以这种方式表示——你能确定哪些数字不能吗？
(来源：New Scientist，BrainTwister #45)
原文：
Set by Alison Kiddle
The number 9 can be written as a difference of two squares as . 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?
上期问题：骰子和卡牌
在每轮游戏中，我们掷三个标准的六面骰子（每个面上的数字为1到6），然后将掷出的数字相加。在长时间的游戏中，总和的平均值（即期望值）是多少？
现在假设我们不是掷骰子，而是从一个编号为1到6的六张卡牌组成的牌堆中抽取三张牌。抽完一张牌后将其放回牌堆并重新洗牌，再抽下一张牌。这种情况下，总和的期望值会发生变化吗？
如果我们改为抽三张牌且不放回（因此概率不再是独立的），那么总和的平均值是多少？
答案：
三个骰子的平均值是10.5。一个骰子上的点数总和是21，将其除以6得出单个骰子的平均值为3.5。骰子是独立的，因此三个骰子点数总和的平均值为3.5 × 3 = 10.5。对于有放回的抽牌，同样的逻辑可以得出答案也是10.5。如果我们在没有放回的情况下抽取三张牌，总共有20种组合（顺序不重要，因为对总和没有影响）。将每种组合的总和（范围从6到15）加起来，再取平均值，也会得出10.5。为了更简单的方法，我们可以注意到在这20种组合的总和中，同样的6个数字（1到6）会重复10次。因此，平均值就是1到6的总和乘以10，再除以20，结果是10.5。
原文:
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.
