Why would a toilet flush completely every other time?

In order for the toilet to flush there has to be a sufficient amount of water in the bowl before flushing.

I will describe the test you need to do to check the toilet.

Remove the tank cover, be sure the tank is full of water to the full mark.

Push the flush handle.

When the rubber flapper flips up, reach down and hold it in the up position until all the water has drained from the tank.

If the toilet flushes completely under this test, but not on it's own, then your flapper is closing too soon, or the tank is not filling as much as it needs to do. The flapper is a real easy replacement, if the tank is not filling, then the fill valve needs to be replaced.

You may have flappers that are made for the new 1.6 gall toilets and need flappers for the 3.6 gallon (older) toilets.

I'd adjust the tension on the flapper valve chain or replace the flapper valves with new ones. Very inexpensive. Is the water level in the tank where it should be?

Check to see if the flapper is worn and needs to be replaced.

Try cleaning out the flow holes under the rim. Insert a wire and wiggle it to dislodge lime or mineral buildup that restricts the water flow into the tank. Clean each hole...there are 10 or so spaced around the rim.

You can also remove the tank lid, flush, then hold the float in its top position when the tank is empty to keep the water from refilling the tank. Open the flapper and pour vinegar or a product such as CLR or LimeAway into the hole under the flapper. DO NOT MIX PRODUCTS. I use about a half gallon of vinegar, wait several minutes, then flush to remove buildup inside the rim. Anything that restricts smooth water flow through the rim interferes with the swirl that is necessary for a complete flush.

I live in an area with very hard water, 30+ grains, and a monthly vinegar treatment keeps our toilets working great. Another tip: Wipe the underside of the flapper and the rim of the tank hole under it to clean off any buildup, ensuring a good seal.