[Math Lair] Black and White, December 28, 1907, page 817: A Christmas Dinner and Some Knotty Problems

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Here are a few Christmas-themed problems by Henry Dudeney. These were published in the newspaper Black and White on December 28, 1907.

A Christmas Dinner and Some Knotty Problems

"Speaking of Christmas puddings," said the Host, as he glanced at the imposing delicacy at the other end of the table, "I am reminded of the fact that a friend gave me a new puzzle the other day respecting one. Here it is," he added, diving into his breast pocket.

"'Problem: To find the contents,' I sup­pose," said the Eton boy.

"No; the proof of that is in the eating. I will read you the conditions."

No. 1.—The Christmas Pudding.

[Christmas pudding]

"'Cut the pudding into two parts, each of exactly the same size and shape, without touch­ing any of the plums. The pudding is to be regarded as a flat disc, not as a sphere.'"

"Why should you regard a Christmas pud­ding as a disc? And why should any reason­able person ever wish to make such an accurate division?" asked the Cynic.

" It is just a puzzle—a problem in dissec­tion. It—it trains the mind, you know."

"I wish some people would mind the trains," whispered the Spinster. Somebody had trodden on hers coming down the stairs.

All in turn had a look at the puzzle, but nobody on that occasion succeeded in solving it. It is a little difficult unless you are ac­quainted with the principle involved in the making of such puddings, but easy enough when you know how it is done.

"Do any of you know how to make dia­monds?" asked the Doctor.

"I wish I did," said the Barrister. "I would then throw up my practice at the Bar, made, as I believe I am, for a very different life."

"What does he say about barmaids? " en­quired the Deaf Lady. "I am very inter­ested in the subject."

"The sort of diamonds I mean," the Doctor explained, ignoring the last question, "may be easily cut from rhomboids."

"What are rhomboids?" several asked at once.

"I had some cut from my froat last year," said the Infant.

"Adenoids, my dear," corrected the Doctor. "A rhomboid is a figure that has its opposite sides equal to one another, but all its sides are not equal nor its angles right angles."

"Speaking of right-handed things," inter­rupted the Widow, who had misunderstood the Doctor's last word, "reminds me of a problem that I was once given and have been trying for years to get somebody to solve for me. Perhaps one of you can help me. 'If a right-handed pair of tongs can pick up a halfpenny turnip in two hours, how long will it take a blackbeetle to crawl to the bottom of a barrel of pitch?'"

"It depends on the quality of the pitch, and the sex of the blackbeetle," said the Eton Boy. " Now, if the turnip is a square root and the——"

" Madam," said the Doctor, with great gravity, "that is what we call an indeter­minate equation—a Diophantine."

"Die of what?" asked the Deaf Lady, but nobody condescended to explain, as all thought it was an intentional attempt at a joke.

No. 2.—Making Diamonds.


The Doctor drew a rhomboid, of which I give an illustration, and declared that it could be cut into two pieces that would fit together and form a perfect diamond. He explained that a perfect diamond or rhombus has all its sides equal but its angles are not right angles.

"In other words," said the incorrigible Eton Boy, "a diamond is an empty square picture frame pulled out of shape by the corners."

"Good!" the Parson broke in. "You remind me of Dr. Johnson's definition of net-work: 'Anything reticulated, with interstices between the intersections,' which a boy improved upon with 'a lot of holes tied together with pieces of string.' I made a new conundrum last week of which I am rather proud."

"What was the text?" asked the Deaf Lady.

"Judges--fourteen--twelve, madam," the Parson replied with ready wit. "'I will now put forth a riddle unto you.'"

After the point had been explained to the Deaf Lady, the Parson gave his new conundrum.

No. 3.—The Parson's Riddle

[Note: Answering this question requires a knowledge of pre-decimal British currency and coins. See pre-decimal British currency for more information on these.]

"What is the difference between the King and the North Star?"

"I didn't know they had quarrelled," said the Spinster.

"Do you want the answer to the nearest foot?" asked the Eton Boy.

"I should have thought," said the Cynic, "that the difference was very obvious. Now, if you had asked, 'What is the resemblance?' one might think it worth consideration."

Still all agreed, when they heard it, that the answer to the conundrum was exceptionally good.

After dinner, the five boys of the household happened to find a parcel of sugar-plums. It was quite unexpected loot, and an exciting scramble ensued, the full details of which I will recount with accuracy, as it forms an interesting puzzle.

No. 4.—The Great Scramble.

You see, Andrew managed to get possession of just two-thirds of a parcel of sugar-plums. Bob at once grabbed three-eights of these, and Charlie managed to seize three-tenths also. Then young David dashed upon the scene, and captured all that Andrew had left, except one-seventh, which Edgar artfully secured for himself by a cunning trick. Now the fun begins in real earnest, for Andrew and Charlie jointly set upon Bob, who stumbled against the fender and dropped half of all that he had, which were equally picked up by David and Edgar, who had crawled under a table and were waiting. Next, Bob sprang at Charlie from a chair, and upset all the latter's collection on to the floor. Of this prize Andrew just got a quarter, Bob gathered up one-third, David got two-sevenths, while Charlie and Edgar divided equally what was left of that stock.

They were just thinking the fray was over when David suddenly struck out in two directions at once, upsetting three-quarters of what Bob and Andrew had last acquired. The two latter, with the greatest difficulty, recovered five-eighths of it in equal shares, but the three others each carried off one-fifth of the same. Every sugar-plum was now accounted for, and they called a truce, and divided equally amongst them the remainder of the parcel. What is the smallest number of sugar-plums there could have been at the start, and what proportion did each boy obtain?

(The answers to the above four Puzzles will be given in our issue of January 11th, 1908.)

Here is the text of the solutions from the January 11th issue:

Solutions (See December 28th Issue).

No. 1.—The Christmas Pudding.

[Christmas pudding solution]

The illustration shows how the pudding may be cut into two parts of exactly the same size and shape. As the principles involved in this puzzle also come into the new one that I am presenting this week, I will not at present do more than remark that the lines must necessarily pass through the points A, B, C, D, and E. But, subject to this condition, they may be varied in an infinite number of ways. For example, at a point midway between A and the edge, the line may be completed in an unlimited number of ways (straight or crooked), provided it be exactly reflected from E to the opposite edge. And similar variations may be introduced at other places.

No. 2.—Making Diamonds.

[rhomboid solution]

By dividing the long sides into four equal parts, and the short sides into three equal parts, as shown, the rhomboid, A, may be cut into two parts that will fit together, as in B, and form the required perfect diamond. If the sides of A measure sixteen inches by nine inches, then it will be seen that if the sixteen loses a fourth and the nine gains a third all sides must be equal.

No. 3.—The Parson's Riddle.

"What is the difference between the King and the North Star?" The answer is, "Nineteen shillings and elevenpence three farthings!" Here we appear to get involved in a second conundrum, "Why is the difference equal to this sum of money?" Well, it is rather obvious that it must be so, since the King is a sovereign and the North Star beyond doubt a far-thing.

No. 4—The Great Scramble.

The smallest number of sugar plums that will fulfil the conditions is 26,880. The five boys obtained respectively: Andrew, 2,863; Bob, 6,335; Charlie, 2,438; David, 10,294; Edgar, 4,950. There was a little trap concealed in the words near the end, "one-fifth of the same," that seemed at first sight to upset the whole account of the affair. But a little thought will show that the words could only mean "one-fifth of five-eights—the fraction last mentioned"; that is, one-eighth of the three-quarters that Bob and Andrew had last acquired.

If you're interested in more Christmas-related puzzles, see How Many Days 'Til Christmas?