When I arrived at Ernie's place last weekend I had some exciting news for him. "did you know", I announced "that a new Kzijekistanian fast-food shop has just opened in town?". "I did", Ernie replied, "because this just arrived in the letter-box", and he held up a glossy flyer that announced:
Spicy Kzijekistanian Filled Flat-Bread Shop
$10.00 per piece
In accordance with regulations of the Kzijekistanian Ethnic Food Committee (KEFOC), each flat-bread is precisely 1.000 m diameter and will be delivered to your home on a square cardboard tray** designed to keep the bread crisp and fresh as possible (Note: dimensions of the tray will be the smallest possible to minimize any over-lapping or under-lapping of your pre-cut*** flat-bread. (Note: Flat-breads will only be placed one layer deep on the tray)
"I ate those when I was visiting Kzijekistan," said Ernie, "very tasty - and $10 seems to be a very fair price too!".
I scanned down the flyer to the small print:
*Free delivery within 3 km of the shop: In accordance with the wishes of the KEFOC our company has made a permanent arrangement with the local Cargo-Bike Appreciation Club. In exchange for bicycle-delivery of flat-breads, the KEFOC will provide free jerseys and accommodation for club members competing at the Annual National Cargo-Bike Olympic Trials. Your orders will help them in training so they can bring home this year's ANCBOT cup.
"That sounds like a noble cause, shall we order one for lunch?", Ernie asked.
I scanned a little further down the flyer:
**In accordance to the demands of the KEFOC, our company has declared itself to be carbon-neutral. In the interests of minimizing packaging waste, there will be a surcharge of one cent per square cm of cardboard tray that is not covered by flat-bread (surcharge rounded down to the nearest cent).
"But that is preposterous", said Ernie, "an un-cut flat-bread would cost an extra $21.46 in packing surcharge! There is no way I would pay that much for one".
I scanned down a little further:
***As directed by existing KEFOC regulations flat-breads can be pre-cut and optimally re-arranged to fit the smallest possible square cardboard packing tray. Each cut**** must be straight, must reach from circumference to circumference, and all cuts must be made before any pieces are moved or re-arranged.
"Well that solves the problem" I replied (feeling very proud of myself), "all we need to do is order a flat-bread with 99 evenly spaced vertical cuts and 99 evenly spaced horizontal cuts. The little 1 cm squares (plus extra edge bits) would certainly fit into a box no more than 90 cm square..." (I did a quick calculation on the back of an envelope) "...so we wouldn't be paying more than $2.46 for the packaging".
"That is a ridiculous solution!" Ernie replied scathingly, "Firstly, I don't want lots of tiny splinters of flat-bread - I want nice big bits wherever practical, so the minimum number of cuts would be preferable, secondly, the packing charges would still be too much, and thirdly you didn't read the final bit of small print."
I scanned down to the end of the document:
****in agreement with KEFOC edicts, the company must charge a cutting levy of one cent per cut.
"So you would be adding an extra $1.98 just in cutting charges" Ernie explained. "To be honest, I wouldn't be happy eating it unless the total cost of extras (packing surcharge plus cutting levy) made up no more than 10% of the total bill." In the end we decided to order pizza instead.
Now I know Ernie does love Kzijekistanian flat-bread and it would be great to surprise him with a home delivery next weekend. But I know he won't be happy if it is too expensive - even if someone else is paying the bill. Can anyone think of a way to cut the bread that will meet with Ernie's requirements?
Hint 1: Looks like a first hint is in order. Ernie and I did manage to find a solution (flat-bread was lovely and tasty), in which the positioning of the pieces of bread in the box had at least two planes of mirror symmetry (when looked at from directly above the box of course).
Hint 2: The round flat-bread, after it has been cut up, but before any pieces have been moved, has exactly the same rotational symmetry and mirror symmetry as it does after the pieces are rearranged and placed in the square box.