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In April, Anna went to the gym 26 times. It is true that there were 6 consecutive days she went to the gym? Repeat but for the case of May.

My attempt: April has 30 days, so she did not go for 4 days. And so 6 consecutive days fit 5 times in 30 days. So we can apply the pigeon hole principle, and notice that there are 5 boxes, and 26 days would be the objects. And so we want to know the average of the number of days per box, and so 26/5 = 5.2. Since we have an average of 5.2, then we would have a box with at least 6 days. So by the pigeon hole principle, we conclude yes.

For May, there are 31 days. Can someone please help me? If I am on the right track. Any suggestion and feedback would really help. Also can someone please help me for the May part.Thank you.

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    no, she might have gone all $26$ times in the same day. – 2017-02-01
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    I don't think the problems assumes that – 2017-02-01
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    so she can only go to the gym once per day? – 2017-02-01
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    The problem does not states . But we can assume that – 2017-02-01
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    You can show that she did not go to the gym in 6 consecutive days by giving a counter-example. Suppose she only goes to the gym at most 5 consecutive days and we group this 5 days as "D". We let "N" represent that she did not go to the gym on that day. Therefore, we will have something like this D-N-D-N-D-N-D-N-D-N-L. This diagram represents that Anna went to the gym for 5 days in a row, then rest for day, and goes to the gym for 5 days in a row, then rest for a day, and repeats. Therefore, in 30 days, she would have went to the gym 25 times and did not go 5 times. Last day = Gym. – 2017-02-01
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    Hence, there can be a possibility that she did not go to the gym in 6 consecutive days! – 2017-02-01
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    But the question is , did she go for 6 consecutive days . So was there a possibility . It turns out if she went for 5 consecutive days and then took a break and then repeat this 4 times , we would have 4(5) + 4+ 6 = 20+ 10= 30 days . Where the 4 means the days she did not go, and the 6 the possibility she did go for 6 consecutive days – 2017-02-01

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