I did make a math error before, so I'll correct it here:

**you'll need to "settle" for 90% confidence**rather than 95%.

n >= ((p*q)/d^2) * Z^2

Assuming accuracy(no dye) = .65 and accuracy(dye) = .90, let's work with the .65, since %'s closer to 50% have larger standard errors, and this will give you a "safer" estimate of the minimum sample size.

Since .90 - .65 = .25, we split the difference to figure out the largest margin of error we can tolerate, which would be approx 12%, so

n >= ((.65 * .35)/.12^2) * 1.282^2

n >= 26

so n = 40 should be enough to detect a significant difference in accuracy at the 90% confidence level

(If I plug in 1.96 for z, using 95% confidence, then n becomes 61, which is, unfortunately, too large)