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The water swirls around and I have to flush several times. There is no suction.
This indicates a siphon problem.
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I have looked for small holes under the toilet rim but can only see fairly large ones at the side and back. Nor can I find any hole at the bottom of the toilet. Do all toilets have one?
Most toilets have rim holes to start the swirling action, How old is this toilet and are you in the USA?
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but the overflow pipe is a polythene tube. I only noticed this yesterday. Is this right?
What is your concern here?
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Nor can I find any hole at the bottom of the toilet. Do all toilets have one?
The newer ones mostly have a jet hole at the bottom of the bowl. These are called siphon jet toilets. The ones with no jet holes are called gravity toilets and the flush action is started by the force of the water from the tank. Make sure the tank level is 1/2" below the top of the overflow tube. Too little water means a week siphon. Can you take a picture of the bowl? Any more information? Back to you, Tom