Answers to Knot 5

Problem. — To mark pictures, giving 3 x’s to 2 or 3, 2 to 4 or 5, and 1 to 9 or 10; also giving 3 o’s to 1 or 2, 2 to 3 or 4, and 1 to 8 or 9; so as to mark the smallest possible number of pictures, and to give them the largest possible number of marks.

Answer. — 10 pictures; 29 marks; arranged thus:

X   X   X   X   X   X   X   X   X   O
X   X   X   X   X       O   O   O   O
X   X   O   O   O   O   O   O   O   O

Solution. — By giving all the x’s possible, putting into brackets the optional ones, we get 10 pictures marked thus:

X   X   X   X   X   X   X   X   X  (X)
X   X   X   X  (X)
X   X  (X)

By then assigning o’s in the same way, beginning at the other end, we get 9 pictures marked thus:

(O)  O
(O)  O   O   O
(O)  O   O   O   O   O   O   O   O

All we have now to do is to run these two wedges as close together as they will go, so as to get the minimum number of pictures — erasing¹ optional marks where by so doing we can run them closer, but otherwise letting them stand. There are 10 necessary marks in the 1st row, and in the 3rd; but only 7 in the end. Hence we erase² all optional marks in the 1st and 3rd rows, but let them stand in the end.

Twenty-two answers have been received. Of these, 11 give no working; so, in accordance with what I announced in my last review of answers, I leave them unnamed, merely mentioning that 5 are right and 6 wrong.

Of the eleven answers with which some working is supplied, 3 are wrong. C. H. begins with the rash assertion that under the given conditions “the sum is impossible. For”, he or she adds (these initialed correspondents are dismally³ vague beings to deal with: perhaps “it” would be a better pronoun), “10 is the least possible number of pictures” (granted): “therefore we must either give 2 x’s to 6, or 2 o’s to 5”. Why “must,” O alphabetical⁴ phantom⁵? It is nowhere ordained⁶ that every picture “must” have 3 marks! Fifee sends a folio page of solution,which deserved a better fate: she offers three answers, in each of which 10 pictures are marked, with 30 marks; in one she gives 2 x’s to 6 pictures; in another to 7; in the third she gives 2 o’s to 5; thus in every case ignoring the conditions. (I pause to remark that the condition “2 x’s to 4 or 5 pictures” can only mean “either to 4 or else to 5”: if, as one competitor holds, it might mean any number not less than 4, the words “or 5” would be superfluous⁷.) I. E. A. (I am happy to say that none of these bloodless phantoms⁸ appear this time in the class-list. Is it Idea with the “D” left out?) gives 2 x’s to 6 pictures. She then takes me to task for using the word “ought” instead of “nought”. No doubt, to one who thus rebels against the rules laid down for her guidance, the word must be distasteful. But does not I. E. A. remember the parallel case of “adder”? That creature was originally “a nadder”: then the two words took to bandying the poor “n”, backwards⁹ and forwards like a shuttlecock, the final state of the game being “an adder”. May not “a nought” have similarly become “an ought” Anyhow, “oughts and crosses” is a very old game. I don’t think I ever heard it called “noughts and crosses”.

In the following Class List, I hope the solitary¹⁰ occupant of III will sheathe¹¹ her claws when she hears how narrow an escape she has had of not being named at all. Her account of the process by which she got the answer is so meagre that, like the nursery tale of “Jack-a-Minory” (I trust I. E. A. will be merciful to the spelling), it is scarcely to be distinguished¹² from “zero”.

Class List

I.

Guy.
Old Cat.
Sea-Breeze.

II.

Ayr.
Bradshaw of the Future.
F. Lee.
H. Vernon.

III.

Cat.

1 erasing   

v.擦掉( erase的现在分词 )；抹去；清除

参考例句：

• He was like a sponge, erasing the past, soaking up the future. 他象一块海绵，挤出过去，吸进未来。 来自辞典例句
• Suddenly, fear overtook longing, erasing memories. 突然，恐惧淹没了渴望，泯灭了回忆。 来自辞典例句

2 erase   

v.擦掉；消除某事物的痕迹

参考例句：

• He tried to erase the idea from his mind.他试图从头脑中抹掉这个想法。
• Please erase my name from the list.请把我的名字从名单上擦去。

3 dismally   

adv.阴暗地，沉闷地

参考例句：

• Fei Little Beard assented dismally. 费小胡子哭丧着脸回答。 来自子夜部分
• He began to howl dismally. 它就凄凉地吠叫起来。 来自辞典例句

4 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。

5 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。

6 ordained   

v.任命（某人）为牧师( ordain的过去式和过去分词 )；授予（某人）圣职；（上帝、法律等）命令；判定

参考例句：

• He was ordained in 1984. 他在一九八四年被任命为牧师。 来自《简明英汉词典》
• He was ordained priest. 他被任命为牧师。 来自辞典例句

7 superfluous   

adj.过多的，过剩的，多余的

参考例句：

• She fined away superfluous matter in the design. 她删去了这图案中多余的东西。
• That request seemed superfluous when I wrote it.我这样写的时候觉得这个请求似乎是多此一举。

8 phantoms   

n.鬼怪，幽灵( phantom的名词复数 )

参考例句：

• They vanished down the stairs like two phantoms. 他们像两个幽灵似的消失在了楼下。 来自辞典例句
• The horrible night that he had passed had left phantoms behind it. 他刚才度过的恐布之夜留下了种种错觉。 来自辞典例句

9 backwards   

adv.往回地，向原处，倒，相反，前后倒置地

参考例句：

• He turned on the light and began to pace backwards and forwards.他打开电灯并开始走来走去。
• All the girls fell over backwards to get the party ready.姑娘们迫不及待地为聚会做准备。

10 solitary   

adj.孤独的，独立的，荒凉的；n.隐士

参考例句：

• I am rather fond of a solitary stroll in the country.我颇喜欢在乡间独自徜徉。
• The castle rises in solitary splendour on the fringe of the desert.这座城堡巍然耸立在沙漠的边际，显得十分壮美。

11 sheathe   

v.（将刀剑）插入鞘；包，覆盖

参考例句：

• Sheathe your swords!把你们的剑插到鞘里！
• The two opponents decided at last to sheathe the sword and met at a dinner given.这两个对手终于决定讲和，在朋友举行的晚宴上会面了。

12 distinguished   

adj.卓越的，杰出的，著名的

参考例句：

• Elephants are distinguished from other animals by their long noses.大象以其长长的鼻子显示出与其他动物的不同。
• A banquet was given in honor of the distinguished guests.宴会是为了向贵宾们致敬而举行的。