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#61 Mark to mark

Set by Katie Steckles

Given a ruler that you know is 6 units long, with a mark 2 units from one end, it is possible to measure distances of 2, 4 and 6 units between some pair of points (using the mark and the ends of the ruler).

Where could you add a second mark to the ruler that would allow you to measure all lengths from 1 to 6 units?

A ruler 11 units in length has a mark at 1. Where can you add two more marks to allow you to measure 10 out of the 11 possible lengths from 1 to 11 units?

Can you find a different way to mark up an 11-unit ruler, using three marks, that allows you to make a different 10 out of the 11 possible lengths?

Solution next week

#60 Say what you see

Solution

There are six ways to arrange three cards. In four of them, at least one card is in its correct position, so the probability is 2/3 = 0.667. With a fourth card, there are 24 ways to shuffle the deck and 15 of these include a match, for a probability of 15/24 = 0.625.

With duplicates, there are two chances for each match, so it is more likely that you will get a match when using this deck. In fact, there are 720 ways to arrange six cards and 455 matches when the cards are all different, for a probability of 0.632, but there are 640 matches with duplicates, for a probability of 0.889.