Ernie had spent most of the afternoon at my place, helping me with the tile grouting in my new bathroom. It had been a difficult and messy job, so when he asked if I could do him a "small favour" I felt I could not refuse.
Ernie went on to explain. "My family are all great fans of the Downton Abbey TV series, and as it is getting into birthday season I thought I would kill several birds with one stone and have a stately home birthday celebration for some of my nieces and nephews. I’ve already organised a catering company to provide appropriately dressed maids and footmen for the occasion, but I thought you might like to act as butler. All you would have to do is dress appropriately and at the right time cut the birthday cakes.” I replied that I would be honoured to accept, so Ernie went on to explain what was required in a little more detail.
"At 10:00am my oldest sister's twins, Fritz and Donaldo, will be celebrating their 10th birthday. I have made them a lovely orange-strawberry short-cake in the shape of a perfect equilateral triangle. As long as they get half each they will be happy." "So just a single cut?" I replied. "A single cut is just right", Ernie replied. "Of course, you need to make sure you keep the Great Aunt happy!" "The Great Aunt?", I asked. "Yes", Ernie explained, "Great Aunt Bismarkia will be arriving with the twins. She was a Time and Motion Manager back in the 1960s, and she always insists on efficiency. She will want you to make sure you use the shortest possible total length of cuts required to divide up the cake appropriately at any cake cutting she witnesses."
"At 11:00am my middle sister's triplets, Maude, Loretta and Gwendolyn, will be celebrating their 11th birthday. I have made them a lovely chocolate cake with mint icing in the shape of a perfect square. It has a small round red marshmallow in the exact middle of the top. Maude and Loretta love marshmallows so will want even shares of the one in the cake, but Gwendolyn hates marshmallows so she will want a piece with no marshmallow. But none of them will be fussed regarding how big or small their piece of cake is." "So just two cuts?" I asked. "Two cuts will be just right", Ernie replied. "Of course, you need to make sure you keep both Great Aunts happy!". "Both Great Aunts!?", I asked. "Yes", Ernie explained, "Great Aunt Lusitania will be arriving with the triplets. She was an Equal Opportunity Advocate back in the 1950s, and she always insists on equality. She will want you to make sure that each of the cuts required to divide up the cake appropriately are of precisely the same length at any cake cutting she witnesses."
"And at 12:00am my youngest sister's quadruplets, Bob, Rob, Jan and Sue will be celebrating their 12th birthday. I have made them a lovely Battenberg cake, with blackberry icing and a liquorish lattice set into the side walls, in the shape of a perfect pentagon. Bob, Rob and Jan love liquorish so they will want equal shares of the perimeter of the cake, while Sue hates liquorish and will want a piece of cake that doesn't touch the perimeter at any point. Oh - and while the first three don't care how big or small their pieces are, Sue will want her piece to be at least half as big as the next larger piece." "So just three cuts?" I asked. "Three cuts will be just right", Ernie replied. "Of course, you need to make sure you keep all three Great Aunts happy!". "Three Great Aunts!!?", I asked. "Yes", Ernie explained, "Great Aunt Titania will be arriving with the quads. She was a Group Theorist back in the 1940s, and she always insists on symmetry. She will want you to make sure that at least one of the final pieces has a mirror symmetry plane at any cake cutting she witnesses."
By now I was feeling a bit concerned – all those rules and requirements. So I made a few more enquiries and Ernie confirmed that there should be no extra pieces of cake cut other than those the children required after the cutting; that each cut had to be straight and could not cut across more than one piece of cake (but a single cut could divide a piece of cake onto one, two or three pieces depending on the shape of the piece and the positions of the previous cuts); that each cake had a side length of exactly 100 mm; and that the old ladies still had great eyesight and could tell if a cut was incorrect by more than a tenth of a degree in angle or a tenth of a mm in length or position. He also confirmed that Great Aunt Bismarkia chooses the shortest-length solution that satisfies all other cake-cutting participants that are present at the time.
Now I’m terrified that I will disappoint some of the children or make the some of the aunts unhappy and ruin the celebrations. Can anyone help me? A set of diagrams showing where and how to cut the cakes would be of great help.