A:  We asked you to think about the problem called tidy boxes. Have you made any progress with it?

翻译：刚才我们请你思考的这个收纳盒问题，你有什么进展吗？

B:  I haven't got very far: perhaps you could begin by putting each colour in its own box. Some colours would have less than ten bricks, so all of that colour would go in one box. But some colours would have more than ten bricks, and we'd have to find a way to use the extra bricks to fill up the gaps in the other boxes. I can't see a way to dothat.

翻译：我目前没有特别大的进展：也许你可以先把每种颜色的积木放进每个颜色相对应的盒子里。有些颜色可能有不到10块积木，所以这些颜色中，所有同色的积木都能够放进它们对应的盒子里。但是有些颜色有10个以上的积木，那样我们就得想办法让多出来的那些积木填补到有空位的那些盒子里。我想不出来怎么做。

A:  OK. Let's think about an easier problem to start with: if we had only one box and ten bricks, all of the same colour, could we solve the problem?

翻译：好。我们先想一个简单点的问题：如果我们只有一个盒子，10块同色积木，我们能解答这道题吗？

B:  Yes, of course: you just put allten bricks in the one box, and it then contains only one colour, so you don't even need the freedom to have two colours in a box.

翻译：当然可以：你只需要把这十块积木都放进一个盒子里。这个盒子只装一种颜色的积木，我们甚至都不需要把条件放宽到一个盒子装两种颜色。

A:  Good. Now suppose you have two boxes and 20 bricks in two different colours. Could you solve the problem then?

翻译：很好。现在假设你有两个盒子，2种颜色的积木20块。你能解答这道题吗？

B:  It's still easy: you can just put the 20 bricks into the two boxes any way you like. Then each box might have a mixture of the two colours, but there are still only two of them.

翻译：依然很简单：你可以把这20块积木随便放进这两个盒子里，这样每个盒子里就可能会混合了两种颜色的积木，但是每个盒子里也只有两个颜色。

A:  So these small versions of the problem are easy. Let's go back to thinking about the original problem, with ten boxes and ten colours of brick. I wonder if, instead of starting to fill all the boxes as you suggested, we could begin by filling just the first box. Can we find a colour with few enough bricks that we can put all of them in the first box?

翻译：所以这个问题的这几个简易版本很容易证明。那我们现在来思考一下原先的问题：十个盒子，十种颜色的积木。我在想，如果我们先试着装第一个盒子，而不是把所有盒子都装上积木，那么我们能否找到一种颜色，这种颜色的积木数目足够少，使得它们都能够被放进第一个盒子里呢？

B:  You mean a colour with ten bricks or less. We can be sure such a colour exists, because the average number of bricks of each colour is ten. If all colours had more than ten bricks, then they would all be above average, and that can't happen.

翻译：你是说找一个积木数目少于或者等于10块的颜色。我们可以肯定这种颜色存在，因为每种颜色平均有10块积木。如果所有颜色的积木数目都超过10块，那么每种颜色的积木平均数就会大于10，这是不可能的。

A:  Right. So we choose a colour that's below average (or equal to the average) and we put all of those bricks into the first box. Now there may be a gap to fill, and I'd like to find another colour with enough bricks to fill the gap. Can I be sure of doing that?

翻译：对。所以我们要选一种颜色，使得这种颜色的积木数目小于（或者等于）每种颜色的积木平均数，然后我们把所有这个颜色的积木都放进第一个盒子里。现在这个盒子里就可能会有空位，然后我想找另外一种颜色，使得这种颜色的积木数目足够填满第一个盒子里的空位。我能确定做到这点吗？

B:  What you could do is to choose a colour with more than the average number of bricks. That colour ought to be enough: the gap to fill must be nine bricks or fewer, and the average is ten.

翻译：你可以找一个颜色，使得这种颜色的积木数目多于积木平均数。这个颜色的积木肯定满足条件，因为如果想要第一个盒子里的空位，需要九块或更少的积木，而每种颜色的积木的平均数是10块。