No Sale

If Chicken McNuggets come in packs of 6, 9, and 20, what’s the largest number of McNuggets that you can’t buy?

Steve Omohundro and Peter Blicher posed this question in MIT Technology Review in May 2002, and Ken Rosato contributed a neat solution.

The answer is 43. To start, notice that we can use the 6-packs and 9-packs to piece together any multiple of 3 other than 3 itself. 43 itself is not divisible by 3, so 6-packs and 9-packs alone won’t get us there, and adding some 20-packs won’t help, since we’d have to add them to a quantity of either 23 or 3, neither of which can be assembled from packs of other sizes. So that shows that 43 itself can’t be reached.

But we still need to show that every larger number can be. Well, we can create all the larger even numbers by adding some quantity of 6-packs to either 36, 38, or 40, and each of those foundations can be assembled from the packs we have (36 = 9 + 9 + 9 + 9, 38 = 20 + 9 + 9, and 40 = 20 + 20). So that takes care of the even numbers. And adding 9 to any of these even numbers will give us any desired odd number above 43, starting with 36 + 9 = 45.

So 43 is the largest number of Chicken McNuggets that can’t be formed by combining 6-packs, 9-packs, and 20-packs.

(I think Henri Picciotto was the first to broach this arresting question, in Games magazine in 1987. Since then, McNuggets have found their way into Happy Meals in 4-piece servings, reducing the largest non–McNugget number to 11. In some countries, though, the 9-piece allotment has been increased to 10 — and in that case there is no largest such number, as no odd quantity can ever be assembled.)

| Science & Math

Earthshapes

https://archive.org/details/earthshapes-portney/

In their 1981 book Facts and Fallacies, Chris Morgan and David Langford note that the biblical reference to the “four corners of the earth” would apply equally well if the world were a tetrahedron.

In a similar spirit, as American airman Joseph Portney was flying over the North Pole in 1968 he wondered, “What if the Earth were … ?” He made sketches of 12 fanciful alternate Earths and gave them to Litton’s Guidance & Control Systems graphic arts group, which created models that were featured in the company’s Pilots and Navigators Calendar of 1969. This made an international sensation, and Portney’s creations were subsequently published for use in classrooms worldwide, inviting students to ponder what life would be like on a cone or a dodecahedron.

Portney graduated from the U.S. Naval Academy and went on to work for Litton on high-altitude navigation problems — for example, designing control systems that could guide an aircraft around one of these strange worlds.

The Internet Archive has the whole complement.

| Science & Math

Bulverism

https://commons.wikimedia.org/wiki/File:C.S.-Lewis.jpg
Image: Wikimedia Commons

Suppose I think, after doing my accounts, that I have a large balance at the bank. And suppose you want to find out whether this belief of mine is ‘wishful thinking.’ You can never come to any conclusion by examining my psychological condition. Your only chance of finding out is to sit down and work through the sum yourself. … It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error.

You must show that a man is wrong before you start explaining why he is wrong. The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became so silly. In the course of the last fifteen years I have found this vice so common that I have had to invent a name for it. I call it ‘Bulverism’. Some day I am going to write the biography of its imaginary inventor, Ezekiel Bulver, whose destiny was determined at the age of five when he heard his mother say to his father — who had been maintaining that two sides of a triangle were together greater than a third — ‘Oh you say that because you are a man.’ ‘At that moment’, E. Bulver assures us, ‘there flashed across my opening mind the great truth that refutation is no necessary part of argument. Assume that your opponent is wrong, and explain his error, and the world will be at your feet. Attempt to prove that he is wrong or (worse still) try to find out whether he is wrong or right, and the national dynamism of our age will thrust you to the wall.’ That is how Bulver became one of the makers of the Twentieth Century.

— C.S. Lewis, “Bulverism: or, The Foundation of Twentieth-Century Thought,” 1941

| Society

Membership

Consider the set (2, 5, 9, 13). Which of these numbers can be tossed out, and for what reason?

We might choose:

  • 2 because it’s the only even number.
  • 9 because it’s the only non-prime.
  • 13 because it doesn’t fit in the sequence An – An-1 = 1 + (An-1 – An-2).

“Hence one could toss out either 2, 9 or 13,” observes Marquette University mathematician George R. Sell. “Therefore one should toss out 5 because it is the only number that cannot be tossed out.”

(George R. Sell, “A Paradox,” Pi Mu Epsilon Journal 2:6 [Spring 1957], 278.)

| Science & Math

“A Matter of Opinion”

https://www.gutenberg.org/cache/epub/52052/pg52052-images.html

A man walks round a pole on the top of which is a monkey. As the man moves, the monkey turns on the top of the pole, so as still to keep face to face with the man. Now, when the man has gone round the pole, has he or has he not gone round the monkey?

— John Scott, The Puzzle King, 1899

| Oddities

A Path-Making Game

Alice and Bob are playing a game. An n×n checkerboard lies between them. Alice begins by marking a corner square, and thereafter the two of them take turns marking squares; each one they choose must be adjacent orthogonally to the last one chosen, so together they’re making a path around the board. When the path can’t continue (because no unmarked adjacent square is available), then the player who moved last wins. For which n can Alice devise a winning strategy? What if she has to start by marking a square adjacent to a corner, rather than the corner itself?

Click for Answer
| Puzzles

Exchange

A Highwayman confronted a Traveler, and covering him with a firearm, shouted: ‘Your money or your life!’

‘My good friend,’ said the Traveler, ‘according to the terms of your demand my money will save my life, my life my money; you imply that you will take one or the other, but not both. If that is what you mean please be good enough to take my life.’

‘That is not what I mean,’ said the Highwayman; ‘you cannot save your money by giving up your life.’

‘Then take it anyhow,’ the Traveler said. ‘If it will not save my money it is good for nothing.’

The Highwayman was so pleased with the Traveler’s philosophy and wit that he took him into partnership and this splendid combination of talent started a newspaper.

— Ambrose Bierce, Fantastic Fables, 1899

| Literature

A Panmagic Geomagic Square

sallows panmagic geomagic square

Another amazing contribution by Lee Sallows:

“The picture above shows a 4×4 geomagic square, which is to say a magic square using geometrical shapes that can be fitted together so as to form an identical target shape, in this case a 4×6 rectangle, rather than numbers adding to a constant sum. In addition, the square is also panmagic, meaning that besides the usual 4 rows, 4 columns, and both main diagonals, the shapes occupying each of the so-called ‘broken’ diagonals, afkn, dejo, cfip, bglm, chin, belo, are also able to tile the rectangle. Lastly, the 4 shapes contained in the corner cells of the four embedded 3×3 sub-squares, acik, bdjl, fhnp, egmo, are also ‘magic’, bringing the total number of target-tiling shape sets to 20, a small improvement over the 16 achieved by a panmagic-only square. With that said, it is worth noting that 4×4 geomagic squares have been found achieving target-tiling scores as high as 48.”

Click the image to expand it. Thanks, Lee!

| Science & Math

The Bebington Puzzle Stones

bebington puzzle stones

Visiting England’s Wirral Peninsula in 1853, Nathaniel Hawthorne came upon a queer battlemented house in the town of Bebington, “quite a novel symbol of decay and neglect,” “probably the whim of some half-crazy person.” “On the wall, close to the street, there were certain eccentric inscriptions cut into slabs of stone, but I could make no sense of them.”

The crazy person was resident Thomas Francis, and the inscriptions had apparently been commissioned to bemuse and entertain passersby. They offer three puzzles. The first presents the image of an inn, The Two Crowns, and the following riddle:

“My name And sign is thirty Shillings just, and he that will tell My Name Shall have a Quart on trust, for why is not Five the Fourth Part of Twenty the Same in All Cases?”

This was easier to guess at the time of its inscription. The landlord of the Two Crowns was Mark Noble, the old English coin known as the noble was worth 6 shillings and eightpence, the mark was worth 13 shillings and fourpence, and two crowns were worth 10 shillings. Together these values total 30 shillings.

The second puzzle is more straightforward: “Subtract 45 From 45 That 45 May Remain.” This seems to refer to the following mathematical curiosity:

  987654321
- 123456789
-----------
  864197532

Each of these figures comprises the digits 1 to 9, so all have the same digit sum: 45.

The last puzzle is the easiest:

  AR
    UBB
I
  NGS
TONEF
  ORAS
 SE
S

Read this straight through and you get A RUBBING STONE FOR ASSES — possibly a comment by Francis on the loiterers who would gather outside his home.

The house was demolished in the 1960s, but the stones can be seen today in the foyer of the library at the Bebington civic center.

| Oddities · Puzzles

“Fifty-Seven to Nothing”

https://books.google.com/books?id=FS8PAAAAQAAJ&pg=PA98

A puzzle by Henry Dudeney:

“It will be seen that we have arranged six cigarettes so as to represent the number 57. The puzzle is to remove any two of them you like (without disturbing any of the others) and so replace them as to represent 0, or nothing.”

Click for Answer
| Puzzles