Amusements in Mathematics, Henry Ernest Dudeney

  • Question 391 - THE MOTOR-CAR RACE

    Sometimes a quite simple statement of fact, if worded in an unfamiliar manner, will cause considerable perplexity. Here is an example, and it will doubtless puzzle some of my more youthful readers just a little. I happened to be at a motor-car race at Brooklands, when one spectator said to another, while a number of cars were whirling round and round the circular track:—

    "There's Gogglesmith—that man in the white car!"

    "Yes, I see," was the reply; "but how many cars are running in this race?"

    Then came this curious rejoinder:—

    "One-third of the cars in front of Gogglesmith added to three-quarters of those behind him will give you the answer."

    Now, can you tell how many cars were running in the race?

  • Question 392 - THE PEBBLE GAME

    Here is an interesting little puzzle game that I used to play with an acquaintance on the beach at Slocomb-on-Sea. Two players place an odd number of pebbles, we will say fifteen, between them. Then each takes in turn one, two, or three pebbles (as he chooses), and the winner is the one who gets the odd number. Thus, if you get seven and your opponent eight, you win. If you get six and he gets nine, he wins. Ought the first or second player to win, and how? When you have settled the question with fifteen pebbles try again with, say, thirteen.

  • Question 393 - THE TWO ROOKS

    This is a puzzle game for two players. Each player has a single rook. The first player places his rook on any square of the board that he may choose to select, and then the second player does the same. They now play in turn, the point of each play being to capture the opponent's rook. But in this game you cannot play through a line of attack without being captured. That is to say, if in the diagram it is Black's turn to play, he cannot move his rook to his king's knight's square, or to his king's rook's square, because he would enter the "line of fire" when passing his king's bishop's square. For the same reason he cannot move to his queen's rook's seventh or eighth squares. Now, the game can never end in a draw. Sooner or later one of the rooks must fall, unless, of course, both players commit the absurdity of not trying to win. The trick of winning is ridiculously simple when you know it. Can you solve the puzzle?

  • Question 394 - PUSS IN THE CORNER

     

    This variation of the last puzzle is also played by two persons. One puts a counter on No. `6`, and the other puts one on No. `55`, and they play alternately by removing the counter to any other number in a line. If your opponent moves at any time on to one of the lines you occupy, or even crosses one of your lines, you immediately capture him and win. We will take an illustrative game.

    A moves from `55` to `52`; B moves from `6` to `13`; A advances to `23`; B goes to `15`; A retreats to `26`; B retreats to `13`; A advances to `21`; B retreats to `2`; A advances to `7`; B goes to `3`; A moves to `6`; B must now go to `4`; A establishes himself at `11`, and B must be captured next move because he is compelled to cross a line on which A stands. Play this over and you will understand the game directly. Now, the puzzle part of the game is this: Which player should win, and how many moves are necessary?

  • Question 395 - A WAR PUZZLE GAME

    Here is another puzzle game. One player, representing the British general, places a counter at B, and the other player, representing the enemy, places his counter at E. The Britisher makes the first advance along one of the roads to the next town, then the enemy moves to one of his nearest towns, and so on in turns, until the British general gets into the same town as the enemy and captures him. Although each must always move along a road to the next town only, and the second player may do his utmost to avoid capture, the British general (as we should suppose, from the analogy of real life) must infallibly win. But how? That is the question.

  • Question 396 - A MATCH MYSTERY

    Here is a little game that is childishly simple in its conditions. But it is well worth investigation.

    Mr. Stubbs pulled a small table between himself and his friend, Mr. Wilson, and took a box of matches, from which he counted out thirty.

    "Here are thirty matches," he said. "I divide them into three unequal heaps. Let me see. We have `14, 11`, and `5`, as it happens. Now, the two players draw alternately any number from any one heap, and he who draws the last match loses the game. That's all! I will play with you, Wilson. I have formed the heaps, so you have the first draw."

    "As I can draw any number," Mr. Wilson said, "suppose I exhibit my usual moderation and take all the `14` heap."

    "That is the worst you could do, for it loses right away. I take `6` from the `11`, leaving two equal heaps of `5`, and to leave two equal heaps is a certain win (with the single exception of `1, 1`), because whatever you do in one heap I can repeat in the other. If you leave `4` in one heap, I leave `4` in the other. If you then leave `2` in one heap, I leave `2` in the other. If you leave only `1` in one heap, then I take all the other heap. If you take all one heap, I take all but one in the other. No, you must never leave two heaps, unless they are equal heaps and more than `1, 1`. Let's begin again."

    "Very well, then," said Mr. Wilson. "I will take `6` from the `14`, and leave you `8, 11, 5`."

    Mr. Stubbs then left `8, 11, 3`; Mr. Wilson, `8, 5, 3`; Mr. Stubbs, `6, 5, 3`; Mr. Wilson,`4, 5, 3`; Mr. Stubbs, `4, 5, 1`; Mr. Wilson, `4, 3, 1`; Mr. Stubbs, `2, 3, 1`; Mr. Wilson, `2, 1, 1`; which Mr. Stubbs reduced to `1, 1, 1`.

    "It is now quite clear that I must win," said Mr. Stubbs, because you must take `1`, and then I take `1`, leaving you the last match. You never had a chance. There are just thirteen different ways in which the matches may be grouped at the start for a certain win. In fact, the groups selected, `14, 11, 5`, are a certain win, because for whatever your opponent may play there is another winning group you can secure, and so on and on down to the last match."

  • Question 397 - THE MONTENEGRIN DICE GAME

    It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation. The two players first select two different pairs of odd numbers (always higher than `3`) and then alternately toss three dice. Whichever first throws the dice so that they add up to one of his selected numbers wins. If they are both successful in two successive throws it is a draw and they try again. For example, one player may select `7` and `15` and the other `5` and `13`. Then if the first player throws so that the three dice add up `7` or `15` he wins, unless the second man gets either `5` or `13` on his throw.

    The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance.

  • Question 398 - THE CIGAR PUZZLE

    I once propounded the following puzzle in a London club, and for a considerable period it absorbed the attention of the members. They could make nothing of it, and considered it quite impossible of solution. And yet, as I shall show, the answer is remarkably simple.

    Two men are seated at a square-topped table. One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another. Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner? The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than `2` feet square and the cigar not more than `4`½ inches long. With those restrictions you may take any dimensions you like. Of course we assume that all the cigars are exactly alike in every respect. Should the first player, or the second player, win?

  • Question 399 - THE TROUBLESOME EIGHT

    Nearly everybody knows that a "magic square" is an arrangement of numbers in the form of a square so that every row, every column, and each of the two long diagonals adds up alike. For example, you would find little difficulty in merely placing a different number in each of the nine cells in the illustration so that the rows, columns, and diagonals shall all add up `15`. And at your first attempt you will probably find that you have an `8` in one of the corners. The puzzle is to construct the magic square, under the same conditions, with the `8` in the position shown.

  • Question 400 - THE MAGIC STRIPS

     

    I happened to have lying on my table a number of strips of cardboard, with numbers printed on them from `1` upwards in numerical order. The idea suddenly came to me, as ideas have a way of unexpectedly coming, to make a little puzzle of this. I wonder whether many readers will arrive at the same solution that I did.

    Take seven strips of cardboard and lay them together as above. Then write on each of them the numbers `1, 2, 3, 4, 5, 6, 7`, as shown, so that the numbers shall form seven rows and seven columns.

    Now, the puzzle is to cut these strips into the fewest possible pieces so that they may be placed together and form a magic square, the seven rows, seven columns, and two diagonals adding up the same number. No figures may be turned upside down or placed on their sides—that is, all the strips must lie in their original direction.

    Of course you could cut each strip into seven separate pieces, each piece containing a number, and the puzzle would then be very easy, but I need hardly say that forty-nine pieces is a long way from being the fewest possible.