Items tagged with conjecture



A geometric construction for the Summer Holiday

Does every plane simple closed curve contain all four vertices of some square?

 This is an old classical conjecture. See:

Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!


SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49)
local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s;
s:=fsolve({ X(a)+X(c) = X(b)+X(d),
            Y(a)+Y(c) = Y(b)+Y(d),
            (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,
            (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},
          {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2});  #lprint(s);
if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi;




X := t->(10-sin(7*t)*exp(-t))*cos(t);
Y := t->(10+sin(6*t))*sin(t);
rng := 0..2*Pi;

proc (t) options operator, arrow; (10-sin(7*t)*exp(-t))*cos(t) end proc


proc (t) options operator, arrow; (10+sin(6*t))*sin(t) end proc


0 .. 2*Pi


s:=SQ(X, Y, rng):
   plot([X,Y,rng], scaling=constrained),
   plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));


Hi everyone, I'm trying to print out Collatz's Conjecture's steps for any given value with the following code but it takes forever and prints nothing. Any idea on how I can get it working ?


checkCollatzValue:=proc(val) local res, remaining;
while res <> 1 do
remaining = irem(val, 2); remaining;
if remaining = 0 then res = val / 2; else res = val * 3 + 1; fi;
end proc;


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