I was pleased to get an invitation to Ernie's place for dinner last Wednesday. The triplets were staying for the week so I knew Ernie would be making an extra effort to cook a gourmet home-cooked feast. So I was surprised when, on arrival, I saw a van from "The Great British Takeaways Company" pulling away from the drive. Still more surprised to find five packets of Fish-n-Chips on the dining-room table. I expressed my concern about the number of calories in the deep-fried food - "I'm on a diet, Ernie". "Ahhh," he replied - "the ideal opportunity to try out my new calorie scanner." Ernie pulled out a small tool that looked something like a miniature blunderbuss and pointed it at one of the packets. "Precisely 625.000 kilo calories," he announced after peering at a small LCD readout. And after convincing me that the scanner could measure any piece of food accurately to the nearest calorie we sat down with the triplets, unwrapped the food, and ate it with our fingers.
The fish was tasty and the chips crisp - all in all a nice meal, but I couldn't help but question why Ernie hadn't produced his usual cordon-bleu repast. "I guess cooking for the girls all week has worn you out?" I suggested. "Oh, to the contrary!" Ernie replied. "Why just this morning I cooked them spicy pork, prawn and coriander dumplings in sweet-and-sour sesame sauce. That was the start of the problem. It was the girls' job to do the washing up afterwards, but they complained that the dishes were extra sticky and dirty. I suggested they should use some of my de-greaser, but unfortunately they miss-heard me and used the de-Gausser instead. It's only designed to work on one screwdriver at a time, so when they loaded it up with all the utensils simultaneously - it blew a coil during the first magnetization cycle. Come and look for yourself."
I followed Ernie into the kitchen and found a great pile of pots, pans, knives, forks, spoons, cleavers, skewers, garlic presses and other cooking paraphernalia all jammed together in the middle of a piece of Ernie-built equipment. I tried to remove a teaspoon only to find it seemingly welded to the saucepan it was sitting next to. "You won't be able to budge it," said Ernie, "the magnetic forces are enormous. We won't be able to untangle anything until I re-wind the burned out coil in the morning. So until then we will have to make do with takeaway food."
At that moment the doorbell rang and Maude went to answer. She was soon back with a large cardboard box (emblazoned Acme Incorporated Tasty Takeaway Treats) which was opened on the table to reveal an enormous cheesecake. The cake was less than a centimeter thick, but almost half a meter in diameter. "Good grief," I couldn't help crying out, "what about my diet? It must be filled with calories!" Ernie pointed the calorie-scanner at the cake and pressed the trigger. "Exactly 4,000 kilo-calories." I decided that a 1/5th share of 800 kilo-calories wouldn't be too much.
The cake came with a set of six disposable plastic sporks, so we would be able to eat it in a civilized manner, but Loretta pointed out a significant problem - "How are we going to divide it into five equal parts without a knife?" We spent a good five minutes searching the kitchen for a suitable implement - every cutting blade seemed to be stuck fast to the de-Gausser, but eventually Ernie found a large circular cookie-cutter. "This should work - aluminium you see - so it didn't get magnetized." He passed the cutter to the triplets and asked them to cut the cake.
"The cutter is the smallest size it could possibly be (relative to the cheesecake)," said Gwendolyn, "to allow us to cut it into exactly four equal-area pieces with exactly two cuts - but I can't see how to divide it for five people." So Ernie took charge and suggested they should cut and divide the cake as follows:
Each triplet would 'choose' a circular part of the cake by making precisely one complete circular cut in the cake with the cookie-cutter.
Each circular cut must pass through the exact centre of the cake.
The cake would not be moved between cuts, but cuts could overlap.
After the cutting:
- Each triplet would get the part or parts that only she chose.
- Ernie would get the part or parts that two or more triplets chose.
- I would get the part or parts that none of the triplets chose.
It took Ernie quite a while to clarify all these instructions so I think the triplets were getting impatient to eat because they didn't make particularly good decisions in the positioning of their cuts.
There were nine pieces (three each for Ernie and me, and one each for the triplets) all of different sizes and shapes. Ernie and Maude appeared to have approximately the right amount (maybe just a bit too much), but Loretta had a much smaller piece than should be her fair share and Gwendolyn's piece was still smaller again. My share was by far the largest - grotesquely too much for me to eat.
I complained that this would totally ruin my diet, so Ernie took charge again. He measured all the pieces with his calorie-scanner and discovered an interesting coincidence.
The ratio of calories in his three pieces combined, compared to the calories in the triplets' three pieces combined, was precisely the same as the ratio of the calories in the triplets' three pieces combined, compared to the calories in my three pieces combined, and this was also precisely the same as the ratio of calories in Gwendolyn's piece, compared to the calories in Loretta's piece.
"Curious," he said, and then suggested that if...
- Ernie kept his two largest pieces,
- Maude kept her piece,
- I cut my largest piece in two and gave the two halves to Loretta and Gwendolyn, and
- I got Ernie's smaller piece plus Loretta's and Gwendolyn's pieces...
...then everyone would have a fair share (within about +/- 90 calories).
Maude pointed out that there was still a problem as it wouldn't be easy to cut my big piece evenly without a knife - at which point Ernie discovered that he had had his pocket-knife in his pocket the whole time! So the cake was soon redistributed fairly (it tasted great) and everyone was happy. Now I just have one small problem. As part of my diet I have to record my calorie intake to the nearest calorie. I can't remember the size of any specific piece, or the coincidental ratio.
Can anyone help me work out just how much cheesecake I got?
Note: In case of confusion, Ernie pointed out that all the cuts were nice and square edged and that the caloric value of each piece of cake was exactly in proportion to its surface area