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A series of vessels of uniform cross section look like letters when viewed face-on. They are 1cm thick, and the corners of the vessels have either whole or half cm values for their coordinates.



Water is poured slowly into each vessel through one of the holes at the top at a rate of 1 cm$^3$ per minute.
Seven of these vessels take the same time to fill up. Which are they?
Which one takes the longest to fill?
Which would fill up first?

The height of the pool of water in one of the vessels is measured and a chart of the height against time is plotted, as follows:



Which vessel does this chart correspond to? Can you explain what each part of the chart corresponds to?

The very observant solver might have noticed that the chart given is actually slightly inaccurate: at certain points it is an over-measurement and at certain points it is an under-measurement. Can you see where and why?

Make charts of height against time for some of the other letters.