For a change of pace, I thought I’d share three interesting mathematical nuggets I’ve run across recently:
Kaprekar’s constant
The first nugget is the magic number 6174, also known as Kaprekar’s constant. Here’s what’s special about it:
1. Take any four-digit number in which at least one digit differs from the rest. For instance, 9998 is OK but 9999 is not. Leading zeros are fine, so 0072 is permitted.
2. Rearrange the digits in descending order. For example, 8953 becomes 9853.
3. Now rearrange the digits in ascending order. 8953 becomes 3589.
4. Subtract the number you got in step 3 from the number you got in step 2. In this case, you get 9853 – 3589 = 6264.
5. Take your new number and perform steps 2-4 on it. You get 6642 – 2466 = 4176.
6. Take that number and repeat steps 2-4.
7. If you keep repeating steps 2-4, you always end up at the number 6174 within at most seven total iterations.
8. Once you arrive at 6174, you stay there forever, because 7641 – 1467 = 6174.
In our example, the entire process looks like this:
Starting number is 8953
9853 – 3589 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174
6174 is the magic number, so we’re done.
Note that since we’re rearranging digits, this result depends on the fact that we’re using decimal arithmetic — base 10. In other bases, there may or may not be a single magic number.
As you’d expect, Kaprekar’s constant was discovered by a guy named Kaprekar — a schoolteacher from Maharashtra. Kaprekar was a weird dude — he checked all 9,990 possibilities by hand. This sort of thing was his hobby, and he would spend hours doing calculations for fun. He said
A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned.
Harmonic series minus 9s
The harmonic series is
![\[ $$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$$ \]](http://theskepticalzone.com/wp/wp-content/ql-cache/quicklatex.com-58ff84d97595196cf9b04be27a01a139_l3.png "Rendered by QuickLaTeX.com")
That series diverges — that is, even though the terms get smaller, the sum is infinite. However, you can turn it into a convergent series by eliminating all terms that contain the digit ‘9’. So 1/9 is excluded, and so are 1/19, 1/29, etc.
At first glance, it feels counterintuitive that this would convert a divergent series into a convergent one. But it makes more sense when you consider the fact that as the denominator increases, the likelihood of finding the digit 9 somewhere in it increases. Only 1 out of 9 single-digit numbers qualifies (9 itself), but 18 out of 100 two-digit numbers qualify (19,29,…89, 90-99), 271 out of 1000 three-digit numbers qualify, and so on.
Without the 9s, the sum is an irrational number approximately equal to 22.92. It has no known relation to any of the fundamental mathematical constants, such as π or e.
The lonely runner problem
Imagine two runners running around a circular track of length one mile. Assume they are running at different speeds and that they continue running forever, as punishment for being MAGA Republicans. Regardless of their speeds, it is guaranteed that at some point they will be exactly opposite each other. That is, the distance between them will be half the length of the track: 1/2 mile.
A “lonely” runner is defined as one who is at least 1/n miles from every other runner, where n is the number of runners. The runners above become lonely when they are opposite each other — 1/2 mile apart — because n = 2, and 1/n is therefore 1/2.
In the case where n = 3, a runner becomes lonely when they are at least 1/3 mile from each of the other two runners. In the case where n = 4, a runner becomes lonely when they are at least 1/4 mile from the other three runners, and so on.
It has been proven that for values of n from 2 through 10, each runner is guaranteed to become lonely at some point. No one knows whether it is true when there are 11 or more runners.
What’s the point of it? Math often doesn’t make sense…
ETA:
Funny thing is when the math makes sense people don’t like it because they don’t understand it …
J-Mac:
Knowledge for the sake of knowledge. I don’t see any practical application of the nuggets in the OP, but they’re interesting. And if you’re like Kaprekar, you can become drunk on numbers. That’s worth something.
If you mean that it can be counterintuitive, I agree. But mathematicians don’t take their results for granted. They demand that the results make sense. That’s why they require formal proofs.
Math is hard.
— Barbie
In the OP, I wrote that eventual loneliness has been proven for all values of n (the number of runners) from 2 through 10. There’s something unsatisfying about that, though, because for simplicity’s sake, we’d also like it to be true for n=1. It’s awkward having a hole there. It’s also counterintuitive to say that a single runner, alone on the track, doesn’t qualify as lonely.
First I considered generalizing the definition of loneliness to include the distance to oneself as well as to all the other runners. If the distance from oneself to oneself is the entire circumference of the track, then this works, and a single runner counts as lonely, because the distance from that runner to him or herself is at least 1/1, or 1 mile. The problem is that it’s more natural to say that the distance from oneself to oneself is zero. “I’m a mile from myself” would be a strange statement to make.
Then it occurred to me that there’s a way to look at this that doesn’t require a change in definitions. Here’s the definition of loneliness again:
In mathematical logic, there is something known as “vacuous truth”, and it applies here. If there is only one runner X, and X therefore has no other runners to be distant from, then the statement “X is at least 1 mile from every other runner” is actually considered to be true. Likewise, the statement “All unicorns are yellow” is also considered to be true, because there are no unicorns to either be or not be yellow. In other words, a statement is considered to be true unless there is at least one counterexample.
That means, sadly, that the lone runner is lonely and will always be lonely, unlike in the n > 1 cases, where the loneliness only lasts for a finite time. But at least we can now say that loneliness is possible for the lone runner, which matches our intuition about what loneliness entails.
Loneliness has therefore been proven inevitable for n=1 through 10.
But now, having written that, it occurs to me that loneliness is guaranteed for n = 0, too, for the same reason. If there are no runners at all, then there are no runners to be lonely, so the statement “Every runner will eventually be lonely” is actually true. But now loneliness doesn’t attach to any runners, because there are no runners for it to attach to. Loneliness is disembodied, a ghostly presence hovering over the track. We might say that the track itself is lonely, if we’re feeling poetic.
With all of that established, we now know that the statement has been proven true for all n such that 0 ≤ n ≤ 10, which is much more satisfying than merely being able to claim proof for 2 ≤ n ≤ 10.
petrushka:
Try telling that to Astrophysicist Barbie.
Another property of Kaprekar’s numbers is that the suitable four-digit numbers (Kaprekar’s numbers) resolve to Kaprekar’s constant in at most seven steps. From 1000 to 9998, there are 356 numbers that resolve to the constant in a single step (besides number 6174, which is the constant itself), 519 numbers in two steps, 2124 numbers in three steps, 1124 numbers in four steps, 1379 numbers in five steps, 1508 numbers in six steps, and 1980 numbers in seven steps. Just for fun, here are the numbers that resolve to the constant in one single step:
1036
1063
1137
1173
1247
1274
1306
1317
1357
1360
1371
1375
1427
1467
1472
1476
1537
1573
1577
1603
1630
1647
1674
1713
1724
1731
1735
1742
1746
1753
1757
1764
1775
2006
2046
2060
2064
2147
2174
2248
2284
2358
2385
2406
2417
2428
2460
2468
2471
2482
2486
2538
2578
2583
2587
2600
2604
2640
2648
2684
2688
2714
2741
2758
2785
2824
2835
2842
2846
2853
2857
2864
2868
2875
2886
3016
3056
3061
3065
3106
3117
3157
3160
3171
3175
3258
3285
3359
3395
3469
3496
3506
3517
3528
3539
3560
3571
3579
3582
3593
3597
3601
3605
3610
3649
3650
3689
3694
3698
3711
3715
3751
3759
3795
3799
3825
3852
3869
3896
3935
3946
3953
3957
3964
3968
3975
3979
3986
3997
4026
4062
4066
4127
4167
4172
4176
4206
4217
4228
4260
4268
4271
4282
4286
4369
4396
4602
4606
4617
4620
4628
4639
4660
4671
4682
4693
4712
4716
4721
4761
4822
4826
4862
4936
4963
5036
5063
5137
5173
5177
5238
5278
5283
5287
5306
5317
5328
5339
5360
5371
5379
5382
5393
5397
5603
5630
5713
5717
5728
5731
5739
5771
5782
5793
5823
5827
5832
5872
5933
5937
5973
6002
6013
6020
6024
6031
6035
6042
6046
6053
6064
6103
6130
6147
6200
6204
6240
6248
6284
6288
6301
6305
6310
6349
6350
6389
6394
6398
6402
6406
6417
6420
6428
6439
6460
6471
6482
6493
6503
6530
6604
6640
6714
6741
6824
6828
6839
6842
6882
6893
6934
6938
6943
6983
7113
7124
7131
7135
7142
7146
7153
7157
7164
7175
7214
7241
7258
7285
7311
7315
7351
7359
7395
7399
7412
7416
7421
7461
7513
7517
7528
7531
7539
7571
7582
7593
7614
7641
7715
7751
7825
7852
7935
7939
7953
7993
8224
8235
8242
8246
8253
8257
8264
8268
8275
8286
8325
8352
8369
8396
8422
8426
8462
8523
8527
8532
8572
8624
8628
8639
8642
8682
8693
8725
8752
8826
8862
8936
8963
9335
9346
9353
9357
9364
9368
9375
9379
9386
9397
9436
9463
9533
9537
9573
9634
9638
9643
9683
9735
9739
9753
9793
9836
9863
9937
9973
Erik:
Yes, I mentioned that in the OP. I should probably also mention why it’s required that at least one of the digits in the starting number be different from the rest. Consider the starting value 6666:
With digits in descending order: 6666
With digits in ascending order: 6666
Subtract ascending from descending: 6666 – 6666 = 0 (or 0000 as a four-digit number)
Now repeat the procedure on 0000:
Descending order: 0000
Ascending order: 0000
Subtract ascending from descending:
0000 – 0000 = 0000
So 0 maps to itself, just like 6174 does. No other numbers do. In mathematical jargon, 0 and 6174 are called the fixed points of the Kaprekar procedure. If all four digits are the same, we end up at the fixed point 0 and stay there. If at least one digit differs, we end up at the fixed point 6174 and stay there. That exhausts the possibilities.
This is not typical behavior. For bases other than 10 and/or digit counts other than 4, there are typically cycles: that is, that is, there are some numbers for which applying the procedure causes you to loop forever. For example, a cycle might look like this: for some number a, applying the procedure repeatedly will cause you to follow a loop: a -> b -> c -> a -> b -> c…
You never stop.
Another mathematical nugget I ran across recently, this one involving random walks:
Imagine an infinite two-dimensional grid. A walker starts at a particular point on the grid. They then take a single step in one of four directions: right, left, up, or down. The direction is chosen randomly. Then they take another step, also chosen randomly. Then another, and another, and another, forever. The steps are always the same length, so that the person is always at the intersection of two grid lines.
The probability that the person will eventually return to the starting point is 100%. That doesn’t mean that it’s certain — this is a quirk due to the fact that there are infinitely many possibilities. The person could in fact go left, left, left… forever and never return to the starting point, despite the fact that the odds of returning are exactly 100%.
If the grid is one-dimensional — a line, in other words — the probability of returning to the starting point is also 100%. But if the grid is three-dimensional, the probability drops down to about 34%.
I don’t understand why. It’s an interesting and counterintuitive result.
I’m surprised or even disappointed…
What is the point of knowledge without purpose?
keiths,
Let’s just assume the evolutionary theory, the origin of life theory, don’t add up.
There is a an Alien, an uncreated being who calls himself God. He can’ have a beginning in the sense we perceive it.
So, what does he do? How does he deal with eternity and immortality?
Are these legitimate questions?
Yes or No?
J-Mac:
I’m surprised and disappointed at your question. Have you never learned something for its own sake, simply out of curiosity, with no application in mind? If not, you’re missing out.
He entertains himself for eternity by pondering mathematical nuggets like the ones discussed in this thread.
Yes, the question of how an immortal being could entertain itself for eternity (assuming it needs entertainment and is capable of experiencing it) is a legitimate question. Why do you ask? It sounds like you may be interested in the answer for its own sake, out of simple curiosity. There’s hope for you yet.
A fourth nugget.
https://www.newsweek.com/rfk-jr-trump-different-way-calculating-drug-prices-warren-senate-11864840
Yes, there is the right way and the wrong way. The right way is that 600 to 10 is a reduction of 98.33% { i.e. (590/600)*100} and the wrong way is anything else.
I mentioned the harmonic series in nugget #2 of the OP:
That sum is infinite, but it grows at a glacially slow pace because the terms get tiny so quickly. Today I learned that if you had started summing that series at the time of the Big Bang, adding one term per second, the total would only be around 40 today.
A cool application of the harmonic series:
The key is that the blocks have to be stacked in a way that takes account of the harmonic series. In the image below, the blocks are all the same length. The overhang of the top block is half the block length, corresponding to the 1/2 term in the series. The next block’s overhang is 1/3, the next is 1/4, etc.
Here’s a real-life image of a stack built with Jenga blocks:
As you can see, the overhangs don’t perfectly match the harmonic series. They’re slightly less. I suppose that’s because it’s really hard to push the blocks out to the tipping point without going over. The surfaces aren’t perfectly level, and there will even be a slight deformation of the lower blocks due to the weight of the stack.
The principle is the same, though, and you can see that the entire length of the top block is beyond the edge of the bottom block.
ETA:
+10 OCD points to anyone who, like me, is annoyed that the Jenga logos aren’t all oriented right side up.
If you believe “vaccines” work, why don’t prove it? You can, can’t you?
J-Mac:
For the same reason I don’t bother proving that atoms exist or that the Big Bang happened.
Not sure why you posted that here when you seem to be responding to this comment in the Antivax thread.