Dudeney 谜题 55

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.

一天晚上，一名伦敦警察看到两辆出租车在可疑的情况下朝相反的方向行驶。这位警官是个特别细心、头脑清醒的人，他掏出皮夹想记下出租车的号码，却发现铅笔丢了。不过幸运的是，他找到了一小块粉笔，用它在附近码头的门口标记了这两个数字。当他回到他的节拍上的同一个位置时，他站起来再次查看数字，并注意到这个奇特之处，所有九个数字(没有零)都被使用并且没有重复的数字，但是如果他将两个数字相乘，它们再次产生九个数字，全部一次，并且仅一次。当一名职员一大早到达码头时，他观察了粉笔痕迹，并小心翼翼地把它们擦掉。由于警察无法记住它们，因此咨询了某些数学家是否有任何已知的方法可以发现所有具有警察注意到的特殊性的数字对；但他们一无所知。然而，这项调查很有趣，并提出了以下问题：哪两个包含所有九位数字的数字相乘时会产生另一个也包含所有九位数字的数字(可能的最高值)？任何地方都不允许有任何事。