Waxwing

Having checked this, you're totally correct. 

Here's a short Python program that checks this empirically.  

import numpy as np

n = 100

b = np.zeros(n)
r = np.ones(n)

for ib in range(n):
   for ir in range(n):
       old_b = b[ib]
       old_r = r[ir]
       new_b = new_r = np.mean([old_b,old_r])
       r[ir] = new_r
       b[ib] = new_b

print(np.mean(b))

A graph of the scaling of the max blue temperature with respect to number of divisions is here:&nbs... (read more)

Treat the blue clay (mass 1, temperature T) as a single lump. Feed it infinitesimal lumps of red clay (mass $dX$, temperature $T_R$). After each infinitesimal feeding, the temperature of the blue clay changes by

$dT=\frac{T+T_{R}dX}{1+dX}-T=-TdX+T_{R}dX$

(Final equality comes from series expansion).

Then you can integrate: ${\int }_0^{T_{final}}\frac{dT}{T_R-T}={\int }_0^{1}dX.$

i.e. $\log (\frac{T_R}{T_R-T_{final}})=1$ 

The final solution is:

$T_{final}=T_R\cdot (1-1/e)$.

It's worth saying that this is the most efficient way to transfer energy from red to blue because each feeding step is thermodynamically reversible.&nbs... (read more)