According to Selfridge and Conway, 3 people’s method is as follows.(See ref for graphical intuition. And refer to wikipedia for more formal description.)

  1. Alice cuts [into what she thinks are thirds].
  2. Betty trims one piece [to create a 2-way tie for largest], and sets the trimmings aside.
  3. Let Chuck pick a piece, then Betty, then Alice. Require Betty to take a trimmed piece if Charlie does not. Call the person who tooked the trimmed piece T, and the other (of Betty and Chuck) NT.
  4. To deal with the trimmings, let NT cut them [into what she thinks are thirds].
  5. Let players pick pieces in this order: T, Alice, then NT.

My questions are

  1. How about to let T cut the trimmings(in step 4. Accordingly, step 5’s order would be NT-Alice-T.)?
  2. How about to change the order of step 5 to T-NT-Alice ?

Are These OK?

UPDATE : I realized why T-NT-Alice order (in step 5) is unfair. In this case, Alice can envy NT.  Then Question 1 remains.

  • $\begingroup$ If T cuts in step 4 and chooses in step 5, it's definitely no longer envy free. $\endgroup$ – Hans Engler Oct 10 '16 at 3:30
  • $\begingroup$ @HansEngler I modified my question. thank you. $\endgroup$ – plhn Oct 10 '16 at 5:39
  • $\begingroup$ If T cuts in step 4, then T should not be the first to choose in step 5. So what should be the order in step 5? There are four possibilities. $\endgroup$ – Hans Engler Oct 10 '16 at 15:41
  • $\begingroup$ @HansEngler I think NT-Alice-T. Is this unfair? $\endgroup$ – plhn Oct 10 '16 at 15:55

Ad 1.

After step 3 Alice thinks, that she nad NT have the same portion of cake.

If we let T to cut trimmings, then in step 5 we have two possible orders (T takes the part as the last one):

  • NT-A-T : Alice see, that NT takes bigger piece than she, so she envy NT.
  • A-NT-T : There is a chance, taht NT won't get the third part of the cake - for example if A cut cake into pieces $30\%, 30\%, 40\%$, B trims $10\%$ from the last part, and T cuts the trimmings into pieces $8\%, 1\%, 1\%$.

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