While not hydroponics, it's possible to grow mushrooms without soil using what is known as a substrate. In this guide, I'll cover everything you need to know about growing mushrooms in a substrate, including how to start growing mushrooms, which substrate is best, as well as the ideal environment for growing mushrooms without soil.
Mushrooms grow and reproduce differently than plants do. The life cycle of mushrooms starts with mushroom spores (sometimes also called mushroom seeds). The spores are dispersed by the wind, and then they eventually settle on soil and other surfaces. When two compatible spores meet, they grow into Mycelium.
Mycelium is a thread-like structure from which mushrooms sprout. Mycelium is the most important requirement when growing mushrooms. In fact, the mushrooms we see with caps and stalks are simply the "fruit" or the reproductive organ of the Mycelium. Unlike plants, mushrooms respire, taking in oxygen and producing carbon dioxide, just like humans.
A substrate is basically a material containing everything the mushroom needs to thrive. There are a couple of ways to grow mushrooms from substrate: from a mushroom kit or from mycelium. Growing mushrooms from a kit is really simple. A mushroom kit is essentially a substrate block with all the nutrients that the mushrooms need. Most importantly, it contains mycelium that will grow as soon as you put the block in chilled and non-chlorinated water and then put them in a dark place with a temperature ranging between 60 and 75°F (16-24°C). Mushrooms grow fast, especially in cold water. You'll be able to harvest your first mushrooms after 3 or 4 days. The biggest downside of mushroom kits is the fact that they aren't infinite. The nutrients will eventually be depleted after 2 or 3 generations of mushrooms, and you won't be able to refill them.
If you want to grow a lot of mushrooms, you can grow them from scratch using spores. Mushroom spores need a growing substrate, which they will populate. The growing substrate is inoculated with spores, and after that, mycelium will begin to develop, fuse, and colonise the substrate. The growing substrate, however, must contain nutrients in some form so that the mycelium can grow. This is because mushrooms can't make their own food, unlike plants. Mushrooms require some form of carbohydrate or sugar to grow and develop. Many people use vermiculite to grow mushrooms similar to its use in hydroponics. Vermiculite is usually mixed with brown rice flour and small amounts of water to form a sterilised cake of sorts on which the mycelium will grow and develop.

To make inoculated vermiculite cakes, you will need vermiculite, brown rice flour, water, mason jars with lids (preferably modified lids with holes for mushroom growing), aluminium foil, micropore tape, a pressure cooker, and a spore syringe or liquid culture. I would suggest using filtered or distilled water to make the cakes. If you decide to use tap water, boil it and cool it down or let it stand in the sun for a day so that the chlorine can evaporate.
Here's how you make inoculated vermiculite cakes:

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