We were wondering last week by the construction of the Gray code in the decimal system. As our "featured user" Manuel Amorós points out:
To achieve the Gray code in base 10, you have to repeat the ordered numbers of the previous bit 10 times, and then prepend each of those 10 images, 10 zeros, 10 ones, 10 doses, etc …
1 bit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
2 bits: (00) (01) (02) …… .. (09) I (19) (18) ……… .. (10) I (20) (21) ………. (29) I ( 39) (38) ……… (30) I (40) (41) …… .. (49) I (59) (58) …… (50) I (60) (61) … .. (69) I (79) (78) … .. (70) I (80) (81) (82) …. (89) I (99) (98) (97) …. (90)
3 bits: etc
That is why Gray himself called "binary code reflected" to the binary version of his code.
And a problem posed by the same reader about a swimming mouse that tries to escape from a cat (see comments of last week) has aroused interesting geometric reflections and has brought up a classic theme of logical riddles: that of the persecutions. Without doubt the most famous is that of Achilles and the turtle, a classic paradox rather than a riddle; but the subject is inexhaustible. Let's see some examples.
Persecutions and estrangements
Let's start with a classic "trickster" to de-numb heated neurons:
In a speed race, you advance to the one that goes in second place; Where do you go after overtaking?
Another simple classic:
A hare carries an initial advantage of 60 of its jumps to a dog. The hare gives 4 jumps while the dog gives 3, but the dog in 5 jumps advances as much as the hare in 8. How many jumps should the dog give to reach the hare?
And an example of the complementary theme of the persecution, which is the distance:
Two cars leave at the same time a roundabout on different rectilinear roads. After one hour a car has traveled 20 km more than the other, and the distance between them exceeds by 20 km the one traveled by the fastest. What can we deduce from these data?
And since we are in time of baths and-unfortunately-water accidents, one of rescue:
A lifeguard is, on land, 5 meters from the edge of the pool. A swimmer asks for help 5 meters from the edge, and the straight line determined by the bather and the lifeguard forms an angle of 45º with the edge. The speed of the lifeguard when running on solid ground is double that of swimming. What do you have to do to get to the swimmer in the shortest possible time? (The lifeguard's route when diving is not taken into account).
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