Off topic: making a tap dance platform questions

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So a very simple project, but I over think and over research. I can’t find the answers to what I’m pondering so figured I’d ask here one there’s some smart guys here.

Making a 4x4 or so tap dancing board for my daughter who made her dance company. It’ll be birch or some hard plywood.

Was thinking of gluing gym mat or something similar to the bottom to give it a little spring and cushion, to protect and give grip to the floor it’s on, and to make it a little quieter if she ever uses it on hardwood.

Would you make the rubber mat underneath be the entire size of the platform? Would you make it in strips and not cover the entire bottom? For any reason, grip, noise, etc, one way make more sense than another?

I’d found many people use simple cheap foam interlocking flooring underneath, like letter block flooring for a little kids play room, but I feel like that’d be slippery if it were ever on hardwood. I don’t want this to be heavy, looking for 1/2 birch plywood as everything says 1/4 is probably too weak, but even though I want it light weight, I thought the interlocking foam tiles wouldn’t be the right choice.

Any advice or considerations appreciated!
 
Real metal "taps" (the metal parts on tap shoes) are what makes the noise not the hardness of the platform.

I'd try framed 2x4's lain on their edges on 12" centers. Sagging between 'studs' will be what you need to avoid.
Stain your hard plywood dance-floor black.
Forget finding a 4X4 hunk of 'Marlight' or Marlee flooring unless you know people who work in display shops.
Like I said, the taps make the noise, not the flooring.

Leave the bottom bare studs IF you find a cushioning material which will loan itself to staying put.
Perhaps using 1/2" plywood squares (12"?) on each corner to provide purchase for said cushioning material would be enough.
Lidl had a blue, "coated foam" exercise mat for ~$10 I cut up and used on a slew of projects.
That stuff would set well on an uneven floor too.

Another idea: I found scraps of vinyl spaghetti floor covering which got used for many projects. This stuff looked like tan spaghetti, randomly lain in a 1/2" layer. The stuff was not rolled so there was no crushing of the material nor any flattening of the spaghetti.
This flooring was used in a hot yoga studio. I've never seen anything like it since.

A friend used some to pad metal seats in his outdoor equipment. It would drain after a rain.
 
There is a platform in this video. Doesn't look like the sturdiest thing. Looks like it's only supported around the perimeter. Maybe it's supposed to flex a lot.

 
They are supposed to be “sprung” floors. I looked into slices of pool noodles to create that spring effect, but I worried if we ever let her use it on hardwood, it’ll slip around. Thought a gym type flooring mat would give the slightest bit of spring while also dampening some of the sound.

I don’t intend to create this with a frame nor to make it make the perfect tap sounds for her. I was choosing birch for its hardiness, because I read pine will quickly get dented up and destroyed by tap shoes.

It’s not that I can’t frame a floor and spend that time and money on it, but it’s more that I’d like to keep it light weight and easily stashed against a wall or behind a couch.
 
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