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Nov 1 at 20:17 comment added Tom Goodwillie Maybe you should first subdivide the simplex barycentrically into $(k+1)!$ congruent pieces, and then choose some subdivision of one of those pieces into simplices that are "reasonably close to being congruent". (And then cut up each of those pieces in the same way.)
Nov 1 at 19:10 history edited LSpice CC BY-SA 4.0
Typos
Nov 1 at 16:52 history asked Neil Strickland CC BY-SA 4.0