Help! Toto Carlyle toilet clogged

We installed a Toto Carlyle one year ago in a remodeled bathroom. It has worked well until it recently became clogged for no apparent or obvious reason. Plunging did not solve the problem. Some brownish water did back up into the bowl after several plunging/flushing attempts. A six foot snake that my husband used did nothing. A roto-rooter guy came out and said that for $250 he would remove the toilet from the floor to investigate. We said no. My child forgot and used the toilet yesterday and it flushed. We thought it had fixed itself. However, the pattern seems to be that it will flush two consecutive times, then act clogged. After several hours, it will flush again for two times, then act clogged. Any ideas of what's causing this and what can be done to fix it? I also smell a faint "sewer" smell. Thanks in advance for the reply.

If you get a couple of flushes and then it acts backed up that tells me the pipe below the toilet is plugged up. Depending on where there is access to this pipe it may be necessary to remove the toilet and snake the line. You need a plumber to do this!

Sounds like you have a partial clog a ways down the pipe. It drains slowly through that area, but there is enough volume to allow a couple of flushes before it backs up to the toilet and there is nowhere for that next flush to go. It's not in the toilet, or it would back up every time. Sounds like it is about 3-4 gallons down the pipe...with a little math, and knowing the diameter of the pipe, you could figure out approximately how far down. 1 gallon = 231 cubic inches. I didn't do the math, but think it is probably further than 6'.

Did you know that every clogged Toto toilet I have seen so far had a clogged waste line below it?

I've never augered a toto yet!

I've never augered a toto yet!

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I've pulled out a toilet paper roller and a GI Joe before.