Try to find out a multi-digit number that if multiplied by the number 9 or any of its multiplications products (18, 27, 36, 45,..) will result in the multiplication factor repeated (n) number of times
The number is: 12345679
12345679 * 9 = 111111111
12345679 * 18 = 222222222
12345679 * 27 = 333333333
And so on...
Charles walks over a railway-bridge. At the moment that he is just ten meters away from the middle of the bridge, he hears a train coming from behind. At that moment, the train, which travels at a speed of 90 km/h, is exactly as far away from the bridge as the bridge measures in length. Without hesitation, Charles rushes straight towards the train to get off the bridge. In this way, he misses the train by just four meters! If Charles would, however, have rushed exactly as fast in the other direction, the train would have hit him eight meters before the end of the bridge.
What is the length of the railway-bridge?
Let the length of the bridge be x meters.
Running towards the train, Charles covers 0.5x-10 meters in the time that the train travels x-4 meters. Running away from the train, Charles covers 0.5x+2 meters in the time that the train travels 2x-8 meters.
Because their speeds are constant, the following holds:
(0.5x-10) / (x-4) = (0.5x+2) / (2x-8)
which can be rewritten to
0.5x2 - 24x + 88 = 0
Using the abc formula we find that x=44, so the railway-bridge has a length of 44 meters.
The Puzzle: Here is a famous prize problem that Sam Loyd issued in 1882, offering $1000 as a prize for the best answer showing how to arrange the seven figures and the eight 'dots' .4.5.6.7.8.9.0. which would add up to 82
The dot over a number signifies that it is a repeater which would go on for ever, as when we endeavor to describe 1/3 decimally as 0.33333 . . . . (etc)
With a series of numbers we place the dot over the first and last, as with 0.97979797979 . . . (etc)
The remarkable feature being that a proper fraction divided by 9s e.g. 46/99 is exactly equal to the numerator with the repeater sign followed by the decimal.