Answers to Knot 2

§ 1. The Dinner Party

Problem. — The Governor of Kgovjni wants to give a very small dinner party, and invites his father’s brother-in-law, his brother’s father-in-law, his father-in-law’s brother, and his brother-in-law’s father. Find the number of guests.

Answer. — One.

A=a
|
-----------
|     |   |
b=B   D=d   C=c
|     |     |
| ------ -----
| |  | | |   |
e=E  | g=G   |
|       |
F=======f

The original diagram was ambiguous, making either D or d the offspring of A. I have amended¹ that. — Ed.

In this genealogy², males are denoted by capitals, and females by small letters.

The Governor is E and his guest is C.

The detailed³ explanation is as follows: His father’s brother-in-law is B or C. His brother’s father-in-law is C. His father-in-law’s brother is C. His brother-in-law’s father is C. Therefore the smallest possible number of guests is 1, C.

Ten answers have been received. Of these, one is wrong, Galanthus Nivalis Major, who insists on inviting⁴ two guests, one being the Governor’s wife’s brother’s father. If she had taken his sister’s husband’s father instead, she would have found it possible to reduce the guests to one.

Of the nine who send right answers, Sea-Breeze is the very faintest breath that ever bore the name! She simply states that the Governor’s uncle might fulfil all the conditions “by intermarriages”! “Wind of the western sea”, you have had a very narrow escape! Be thankful to appear in the Class List at all! Bog-Oak and Bradshaw of the Future use genealogies⁵ which require 16 people instead of 14, by inviting the Governor’s father’s sister’s husband instead of his father’s wife’s brother. I cannot think this so good a solution as one that requires only 14. Caius and Valentine deserve special mention as the only two who have supplied genealogies.

Class List.

I.

Bee.
Matthew Matticks.
Caius.
Old Cat.
M. M.
Valentine.

II.

Bog-Oak.
Bradshaw of the Future.

III.

Sea-Breeze.

§ 2. The Lodgings⁶

Problem. — A Square has 20 doors on each side, which contains 21 equal parts. They are numbered all round, beginning at one corner. From which of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the other three, least?

Answer. — From No. 9.

[the arrangement of the square]

Let A be No. 9, B No. 25, C No. 52, and D No. 73. Then

AB = √(12² + 5²) = √169 = 13;
AC =21;
AD = √(9² + 8²) = √145 = 12 +
(N.B. i.e, “between 12 and 13”)
BC = √(16² + 12²) = √400 = 20;
BD = √(3² + 21²) = √450 = 21 +;
CD = √(9² + 13²) = √250 = 15 +;

Hence the sum of distances from A is between 46 and 47; from B, between 54 and 55; from C, between 56 and 57; from D, between 48 and 51. (Why not “between 48 and 49”? Make this out for yourselves.) Hence the sum is least for A.

Twenth-five solutions have been received. Of these, 15 must be marked “zero”, 5 are partly right, and 5 right’. Of the 15, I may dismiss Alphabetical⁷ Phantom⁸, Bog-Oak, Dinah Mite⁹, Fifee, Galanthus Nivalis Major (I fear the cold spring has blighted¹⁰ our Snowdrop), Guy, H.M.S. Pinafore, Janet, and Valentine with the simple remark that they insist on the unfortunate lodgers¹¹ keeping to the pavement. (I used the words “crossed to Number Seventy-three” for the special purpose of showng that short cuts were possible.) Sea-Breeze does the same, and adds that “the result would be the same” even if they crossed the Square, but gives no proof of this. M. M. draws a diagram, and says that No. 9 is the house, “as the diagram shows”. I cannot see how it does so. Old Cat assumes that the house must be No. 9 or No. 73. She does not explain how she estimates the distances. Bee’s arithmetic is faulty: she makes √169+√442+√130=741. (I suppose you mean √742, which would be a little nearer the truth. But roots cannot be added in this manner. Do you think √9 + √16 is 25, or even √25!) But Ayr’s state is more perilous¹² still: she draws illogical conclusions with a frightful¹³ calmness. After pointing out (rightly) that AC is less than BD, she says, “therefore the nearest house to the other three must be A or C.” And again, after pointing out (rightly) that B and D are both within the half-square containing A, she says, “therefore” AB +AD must be less than BC + CD. (There is no logical force in either “therefore”. For the first, try Nos. 1, 21, 60, 70: this will make your premiss true, and your conclusion false. Similarly, for the second, try Nos. 1, 30, 91, 71.)

Of the five partly-right solutions, Rags and Tatters and Mad Hatter (who send one answer between them) make No. 25 6 units from the corner instead of 5. Cheam, E. R. D. L., and Meggy Potts leave openings at the corner’s of the square, which are not in the data: moreover Cheam gives values for the distances without any hint that they are only approximations, Crophi and Mophi make the bold and unfounded assumption that there were really 21 houses on each side, instead of 20 as stated by Balbus. “We may assume”, they add, “that the doors of Nos. 21, 42, 63, 84, are invisible from the centre of the Square”! What is there, I wonder, that Crophi and Mophi would not assume?

Of the five who are wholly right, I think Bradshaw of the Future, Caius, Clifton C., and Martreb deserve special praise for their full analytical¹⁴ solutions. Matthew Matticks picks out No. 9, and proves it to be the right house in two ways, very neatly¹⁵ and ingeniously, but why he picks it out does not appear. It is an excellent synthetical¹⁶ proof, but lacks the analysis which the other four supply.

Class List.

I.

Bradshaw of the Future.
Clifton C.
Caius.
Martreb.

II.

Matthew Matticks.

III.

Cheam.
Meggy Potts.
Crophi and Mophi.
Rags and Tatters.
E. R. D. L.
Mad Hatter.

A remonstrance¹⁷ has reached me from Scrutator on the subject of Knot 1, which he declares was “no problem at all”. “Two questions”, he says, “are put. To solve one there is no data: the other answers itself.” As to the first point, Scrutator is mistaken; there are (not “is”) data sufficient to answer the question. As to the other, it is interesting to know that the question “answers itself”, and I am sure it does the question great credit: still I fear I cannot enter it on the list of winners, as this competition is only open to human beings.

1 Amended   

adj. 修正的 动词amend的过去式和过去分词

参考例句：

• He asked to see the amended version. 他要求看修订本。
• He amended his speech by making some additions and deletions. 他对讲稿作了些增删修改。

2 genealogy   

n.家系，宗谱

参考例句：

• He had sat and repeated his family's genealogy to her,twenty minutes of nonstop names.他坐下又给她细数了一遍他家族的家谱，20分钟内说出了一连串的名字。
• He was proficient in all questions of genealogy.他非常精通所有家谱的问题。

3 detailed   

adj.详细的，详尽的，极注意细节的，完全的

参考例句：

• He had made a detailed study of the terrain.他对地形作了缜密的研究。
• A detailed list of our publications is available on request.我们的出版物有一份详细的目录备索。

4 inviting   

adj.诱人的，引人注目的

参考例句：

• An inviting smell of coffee wafted into the room.一股诱人的咖啡香味飘进了房间。
• The kitchen smelled warm and inviting and blessedly familiar.这间厨房的味道温暖诱人，使人感到亲切温馨。

5 genealogies   

n.系谱，家系，宗谱( genealogy的名词复数 )

参考例句：

• Tracing back our genealogies, I found he was a kinsman of mine. 转弯抹角算起来——他算是我的一个亲戚。 来自汉英文学 - 中国现代小说
• The insertion of these genealogies is the more peculiar and unreasonable. 这些系谱的掺入是更为离奇和无理的。 来自辞典例句

6 lodgings   

n. 出租的房舍, 寄宿舍

参考例句：

• When he reached his lodgings the sun had set. 他到达公寓房间时，太阳已下山了。
• I'm on the hunt for lodgings. 我正在寻找住所。

7 alphabetical   

adj.字母（表）的，依字母顺序的

参考例句：

• Please arrange these books in alphabetical order.请把这些书按字母顺序整理一下。
• There is no need to maintain a strict alphabetical sequence.不必保持严格的字顺。

8 phantom   

n.幻影，虚位，幽灵；adj.错觉的，幻影的，幽灵的

参考例句：

• I found myself staring at her as if she were a phantom.我发现自己瞪大眼睛看着她，好像她是一个幽灵。
• He is only a phantom of a king.他只是有名无实的国王。

9 mite   

n.极小的东西；小铜币

参考例句：

• The poor mite was so ill.可怜的孩子病得这么重。
• He is a mite taller than I.他比我高一点点。

10 blighted   

adj.枯萎的，摧毁的

参考例句：

• Blighted stems often canker.有病的茎往往溃烂。
• She threw away a blighted rose.她把枯萎的玫瑰花扔掉了。

11 lodgers   

n.房客，租住者( lodger的名词复数 )

参考例句：

• He takes in lodgers. 他招收房客。 来自《简明英汉词典》
• A good proportion of my lodgers is connected with the theaters. 住客里面有不少人是跟戏院子有往来的。 来自辞典例句

12 perilous   

adj.危险的，冒险的

参考例句：

• The journey through the jungle was perilous.穿过丛林的旅行充满了危险。
• We have been carried in safety through a perilous crisis.历经一连串危机，我们如今已安然无恙。

13 frightful   

adj.可怕的；讨厌的

参考例句：

• How frightful to have a husband who snores!有一个发鼾声的丈夫多讨厌啊！
• We're having frightful weather these days.这几天天气坏极了。

14 analytical   

adj.分析的；用分析法的

参考例句：

• I have an analytical approach to every survey.对每项调查我都采用分析方法。
• As a result,analytical data obtained by analysts were often in disagreement.结果各个分析家所得的分析数据常常不一致。

15 neatly   

adv.整洁地，干净地，灵巧地，熟练地

参考例句：

• Sailors know how to wind up a long rope neatly.水手们知道怎样把一条大绳利落地缠好。
• The child's dress is neatly gathered at the neck.那孩子的衣服在领口处打着整齐的皱褶。

16 synthetical   

adj.综合的，合成的

参考例句：

• Studying the value of science is a new and synthetical study. 科学价值的向度研究是一个崭新的综合性跨学科领域研究。 来自互联网
• Among them, sea-island structure flexibilizer had good synthetical properties. 相比较而言，端环氧基聚氨酯醚海岛结构增韧剂的综合性能较好。 来自互联网

17 remonstrance   

n抗议，抱怨

参考例句：

• She had abandoned all attempts at remonstrance with Thomas.她已经放弃了一切劝戒托马斯的尝试。
• Mrs. Peniston was at the moment inaccessible to remonstrance.目前彭尼斯顿太太没功夫听她告状。