Amusements in Mathematics, Henry Ernest Dudeney

  • Question 81 - THE NINE COUNTERS

    I have nine counters, each bearing one of the nine digits, `1, 2, 3, 4, 5, 6, 7, 8` and `9`. I arranged them on the table in two groups, as shown in the illustration, so as to form two multiplication sums, and found that both sums gave the same product. You will find that `158` multiplied by `23` is `3,634`, and that `79` multiplied by `46` is also `3,634`. Now, the puzzle I propose is to rearrange the counters so as to get as large a product as possible. What is the best way of placing them? Remember both groups must multiply to the same amount, and there must be three counters multiplied by two in one case, and two multiplied by two counters in the other, just as at present.

  • Question 82 - THE TEN COUNTERS

    In this case we use the nought in addition to the `1, 2, 3, 4, 5, 6, 7, 8, 9`. The puzzle is, as in the last case, so to arrange the ten counters that the products of the two multiplications shall be the same, and you may here have one or more figures in the multiplier, as you choose. The above is a very easy feat; but it is also required to find the two arrangements giving pairs of the highest and lowest products possible. Of course every counter must be used, and the cipher may not be placed to the left of a row of figures where it would have no effect. Vulgar fractions or decimals are not allowed.

  • Question 83 - DIGITAL MULTIPLICATION

    Here is another entertaining problem with the nine digits, the nought being excluded. Using each figure once, and only once, we can form two multiplication sums that have the same product, and this may be done in many ways. For example, 7x658 and 14x329 contain all the digits once, and the product in each case is the same—`4,606`. Now, it will be seen that the sum of the digits in the product is `16`, which is neither the highest nor the lowest sum so obtainable. Can you find the solution of the problem that gives the lowest possible sum of digits in the common product? Also that which gives the highest possible sum?

  • Question 84 - THE PIERROT'S PUZZLE

    The Pierrot in the illustration is standing in a posture that represents the sign of multiplication. He is indicating the peculiar fact that `15` multiplied by `93` produces exactly the same figures (`1,395`), differently arranged. The puzzle is to take any four digits you like (all different) and similarly arrange them so that the number formed on one side of the Pierrot when multiplied by the number on the other side shall produce the same figures. There are very few ways of doing it, and I shall give all the cases possible. Can you find them all? You are allowed to put two figures on each side of the Pierrot as in the example shown, or to place a single figure on one side and three figures on the other. If we only used three digits instead of four, the only possible ways are these: `3` multiplied by `51` equals `153`, and `6` multiplied by `21` equals `126`.

  • Question 85 - THE CAB NUMBERS

    A London policeman one night saw two cabs drive off in opposite directions under suspicious circumstances. This officer was a particularly careful and wide-awake man, and he took out his pocket-book to make an entry of the numbers of the cabs, but discovered that he had lost his pencil. Luckily, however, he found a small piece of chalk, with which he marked the two numbers on the gateway of a wharf close by. When he returned to the same spot on his beat he stood and looked again at the numbers, and noticed this peculiarity, that all the nine digits (no nought) were used and that no figure was repeated, but that if he multiplied the two numbers together they again produced the nine digits, all once, and once only. When one of the clerks arrived at the wharf in the early morning, he observed the chalk marks and carefully rubbed them out. As the policeman could not remember them, certain mathematicians were then consulted as to whether there was any known method for discovering all the pairs of numbers that have the peculiarity that the officer had noticed; but they knew of none. The investigation, however, was interesting, and the following question out of many was proposed: What two numbers, containing together all the nine digits, will, when multiplied together, produce another number (the highest possible) containing also all the nine digits? The nought is not allowed anywhere.

  • Question 86 - QUEER MULTIPLICATION

    If I multiply `51,249,876` by `3` (thus using all the nine digits once, and once only), I get `153,749,628` (which again contains all the nine digits once). Similarly, if I multiply `16,583,742` by `9` the result is `149,253,678`, where in each case all the nine digits are used. Now, take `6` as your multiplier and try to arrange the remaining eight digits so as to produce by multiplication a number containing all nine once, and once only. You will find it far from easy, but it can be done.
    Topics:
    Arithmetic

  • Question 87 - THE NUMBER CHECKS PUZZLE

    Where a large number of workmen are employed on a building it is customary to provide every man with a little disc bearing his number. These are hung on a board by the men as they arrive, and serve as a check on punctuality. Now, I once noticed a foreman remove a number of these checks from his board and place them on a split-ring which he carried in his pocket. This at once gave me the idea for a good puzzle. In fact, I will confide to my readers that this is just how ideas for puzzles arise. You cannot really create an idea: it happens—and you have to be on the alert to seize it when it does so happen. It will be seen from the illustration that there are ten of these checks on a ring, numbered `1` to `9` and `0`. The puzzle is to divide them into three groups without taking any off the ring, so that the first group multiplied by the second makes the third group. For example, we can divide them into the three groups, `2`—`8` `9` `7`—`1` `5` `4` `6` `3`, by bringing the `6` and the `3` round to the `4`, but unfortunately the first two when multiplied together do not make the third. Can you separate them correctly? Of course you may have as many of the checks as you like in any group. The puzzle calls for some ingenuity, unless you have the luck to hit on the answer by chance.

  • Question 88 - DIGITAL DIVISION

    It is another good puzzle so to arrange the nine digits (the nought excluded) into two groups so that one group when divided by the other produces a given number without remainder. For example, `1` `3` `4` `5` `8` divided by `6` `7` `2` `9` gives `2`. Can the reader find similar arrangements producing `3, 4, 5, 6, 7, 8`, and `9` respectively? Also, can he find the pairs of smallest possible numbers in each case? Thus, `1` `4` `6` `5` `8` divided by `7` `3` `2` `9` is just as correct for `2` as the other example we have given, but the numbers are higher.

  • Question 89 - ADDING THE DIGITS

    If I write the sum of money, £`987, 5`s. `4`½d.., and add up the digits, they sum to `36`. No digit has thus been used a second time in the amount or addition. This is the largest amount possible under the conditions. Now find the smallest possible amount, pounds, shillings, pence, and farthings being all represented. You need not use more of the nine digits than you choose, but no digit may be repeated throughout. The nought is not allowed.

  • Question 90 - THE CENTURY PUZZLE

    Can you write `100` in the form of a mixed number, using all the nine digits once, and only once? The late distinguished French mathematician, Edouard Lucas, found seven different ways of doing it, and expressed his doubts as to there being any other ways. As a matter of fact there are just eleven ways and no more. Here is one of them, `91 5742/638`. Nine of the other ways have similarly two figures in the integral part of the number, but the eleventh expression has only one figure there. Can the reader find this last form?