Enigma 1131: Numbers in boxes
23 January 2017
Posted by on
From New Scientist #2287, 21st April 2001 [link]
I have a row of boxes numbered 1, 2, 3, …, in order, going on forever. Each box contains a piece of paper on which is written a positive fraction, for example box 1 contains 2/5. When I looked at the numbers in all the boxes I found the following was true:
If you chose any positive fraction then I can find a particular box so that all the numbers in the boxes after that particular box will be less than your fraction.
For example, if you choose 1/3 then all the numbers in the boxes after box 44 are less than 1/3.
This morning I was looking for 3 boxes containing numbers adding up to 1. In fact I made a list, L, of all such three’s of boxes.
Question: Which of the following statements can you say for certain are true?
(a) All the boxes on list L come before box 100;
(b) All the boxes on list L come before box 1,000,000;
(c) I can find a particular box so that all the boxes on list L come before my particular box.