Bad Arrangements in Modfiveland
The numbers 1, 2, 3, 4, and 5 are to be arranged in a circle. They are the only numbers in Modfiveland, therefore 1 and 5 are considered to be consecutive integers. An arrangement is considered
bad
if no consecutive integers are consecutive (in that particular arrangement). How many
bad
arrangements exist if reflections and rotations are not considered distinct arrangements?
The answer is 1.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
When first viewing this problem, it would appear to be somewhat complex. It isn't. Start by noting that shifting the digits over places doesn't change the problem (e.g. You may think that 12345 is different from 23451, but it isn't because it is actually a rotation of 12345). Also note that reversing an ordering is also the same thing (12345 is the same as 54321). Notice how you may start always with the same digit, because it doesn't matter. Let's say 1 is the first digit. Then the only numbers that can follow are either a 3 or a 4 (remember, 5 is considered to be consecutive to 1). Case 1: The first 2 digits are 13. By using simple casework, we see that the only way to do this with 13 is 13524. Case 2: The first 2 digits are 14. We see that the only way to do this is 14253. It would appear that the answer is 2, but look again. Remember that flipping the number's orientation is considered the same. Notice what 14253 is when it is reversed: 35241. Notice now that if this is shifted over by 1 to the right that this is equal to 13524. Deja vu? That was the first number we got. Thus there is only 1 value that works.