Sunday, December 24, 2006

The mysterious 6174 revisited

From my earlier post on 6174 comes a simpler 3 digit variant. Select 3 integers which are not all the same. Arrange these digits to form the largest and smallest numbers and subtract them. Repeat this process with the result. For instance,

981-189=792
972-279=693
963-369=594
954-459=495

You will end up with 495!
The following gives a proof that all 3 digit numbers which are not 0,111,222,333,...,999 will end up with 495. This proof can be adapted to 4 digits as well, as in the case of 6174.

Let x,y,z be 3 digits such that z is the largest while x is the smallest, and x,y and z are not all equal. Then if we subtract the largest number formed from the smallest number formed, we have

zyx
- xyz
ABC

Since x < z
C=10+x-z,

Also,

B=10+y-1-y

And

A=z-1-x

Hence B = 9 = z since z is the largest digit. We want A and C to be either x or y. Solving, we see that x=4 and y=5. Hence 495 is the only number that repeats itself after an iteration.

But do any set of 3 randomly selected integers not all equal will lead to 495 after a finite number of iterations? First we see that for x,y and z,

z(100)+y(10)+x-(x(100)-y(10)-z)=(z-x)(99)

Hence the resulting number after any iteration is a 3-digit multiple of 99.
Hence we only need to check 9 possible numbers: 99,198,297,296,495,594,693,792 and 891 (10(99)=990 which is the same as 99)

Since all the above numbers lead to 495, QED.

Surprisingly, 495=5(99) which is in the middle of the pack of multiples of 99!

Can you see if you can generalise this result to 5 digit numbers and above?

Many drivers make the assumption that braking distance would be directly proportional to their speed. We shall investigate this by using some simple quadratic equations.

From our kinematic equation

v^2=u^2+2as,

we want to estimate our braking distance d, hence v=0. If we include reaction time t, we get

d=u^2/2a+ut

Assume a= 10m/s/s and t = 0.75s for an average driver. Then we have

d=u^2/20+(0.75)u

If a vehicle speeds at 100km/h (27.8m/s), d=59.5m

If a vehicle slows down to half that speed, ie 50 km/h (13.9m/s), d=20.1 m, which is roughly one third of the original distance, and not half of the latter!

Quadratic equations apply to our everyday lives, and this being one example. Students can be given tasks such as investigating how reaction time affects the braking distance, and why it is important for drivers to slow down in congested areas, and why drivers should never tail gate or use their handphones while driving (distractions can increase their reaction times to 3s or more.)

More can be found in this link:
http://www.science.org.au/nova/058/058key.htm

Saturday, December 16, 2006

The mysterious number 6174

Think of any 4 non-zero digits, where not all four digits are the same. Arrange the digits in descending order and ascending order and compute the difference. For instance, if the digits are 3,3,2 and 1,

3321-1233=2088

Repeat the above step by rearranging the result into ascending and descending order and compute the difference.

8820-0288=8532

Reiterating,

8532-2358=6174,

Strangely,

7641-1467=6174

The process terminates when the number reaches 6174. We can use another example,

9750-0579=9171
9711-1179=8532
8532-2358=6174

and Viola! We get the same result again! Can you figure out why this happens?