Google researcher, long out of math, cracks devilish problem about sets
quantamagazine.org
quantamagazine.org
A related toy example that comes to mind is the puzzle where you're given 12 metal balls and told that one of them is either heavier or lighter than the rest. You're then given a balance/scale and instructed to figure out which ball is different and whether it is heavier or lighter using the balance a maximum of three times to weigh different combinations of balls.
Part of the reason this is solvable is because there are 12x2=24 different possibilities (or log2(24) = 4.58 bits) for the ball ID/weight and 27=3x3x3 possible outcomes (or log2(27) = 4.75 bits) for the weighing process (left heavier, right heavier, or equal in each use). Thus, there's a chance the weighing sequence could convey the ball ID/weight. If there were 14 balls, however, then you would require log(28) = 4.81 bits of information to specify the ball ID/weight, so the puzzle would be unsolvable, since there wouldn't be enough information available in the weighing sequence to specify the ball ID/weight.
It’s also fun to scale the problem from 3 balls up to see how the solution evolves.
> The 12-ball problem is a fun one and IMO the key revelation is that you do not need to determine whether the ball is lighter or heavier (which isn’t possible anyway).
Huh?
I'm confused how the response to "you must identify whether the odd ball is heavier or lighter" can be "the key to this is realizing you don't need to know whether the odd ball is heavier or lighter, which is good because that's impossible anyway".
The usual solution uses the fact there are 3 places to put items: left scale, off scale, and right scale. Then those solutions do if/then cases to juggle items around to find the mis-weighted ball.
My solution numbered the balls in base 3, then did four weighings with no if/thens,. The weightings each gave a base 3 result: left, balance, right. This formed the digits of the mis-weighted ball.
The idea is to make an error correcting code in base 3, and use the "weighings" as the correction matrix to isolate the bad "bit".
This idea generalizes to solve a massive range of find the odd item(s) problems.
Then 3 weighings, 3 choices per, gives 27 possible outcomes, which is enough to distinguish which of 12 items is off, as well as whether or not it's higher or lower in weight (another factor of 2), for 24 choices. So there is a little extra in the 3=27-24 information theoretical bounds.
Using this idea, you can generate all sorts of hard coin problems, vastly harder than this one :), by mining error correction tables like http://www.codetables.de/ and converting to word problems.
If you like cool problems, here is one I have not found a simple solution for, but I once saw it in a place where a simple solution should exist.
You have an odd number >=3 of items with the property that any time you remove one, there is a way to split the remaining items into two equal count piles of equal weight. Show all items have the same weight.
It's easy for 3 items, A,B,C. Remove C, A balances B, then do A,C, now all equal.
It's easy to check by hand for a few small cases.
I proved it using more advanced stuff from lin alg. But I've never found a simple solution yet.
EDIT: corrected with "equal count piles"
For number 5, I can use items of weights 3 1 1 1 1 and they are not equal. If you remove 3, you get 1+1=1+1 and in case you remove only of 1s, you can split 3 = 1+1+1.
But no such proof can exist because the claim is not true. When you posted your comment, Fetiorin had already given a counterexample.
The puzzle is related to the game 20 questions. If you can ask only 20 yes/no questions, you can differentiate 2^20 items. For example, with 20 yes/no questions, you can determine an integer in 1 to 1,000,000 (technically, up to 2^20).
Now consider you need to figure out what integer between 1 and 1,000,000 I am think of, but I am allowed to lie at most once. Now how many questions will you need to ensure you get the answer?
Doing this by hand and creating more and more convoluted tricks is possible, but in reality this is another error correction problem: The original problem is a binary message of length 20 bits, from which you get the 20 bit message.
With at most one lie, there is a 20 bit message, me (the noisy channel) can add at most one error, so you need the minimal length binary code that takes a message of length 20 and a max error of 1 bit.
To find the number needed, use a coding table like http://www.codetables.de/.
So you need to find how many more bits you need. To fix d bits, you need 2d+1 'distance' in your code to make sure d errors do not move one valid point within Hamming distance d of another answer, so we need d = 3.
So you need a binary code, d = 3, and encoding a message of length k=20. Click on Linear Codes, then the GF2 button, and you see the big table of best codes. Find the column with k = 20 (our message length), scroll down over increasing n (number of questions needed), until the first d=3 entry. There n = 25, so 25 questions will do it. This is the result http://www.codetables.de/BKLC/BKLC.php?q=2&n=25&k=20
This is provably best, otherwise the table would have a range if there were not proof (look around the table, notice some are ranges).
To design the questions, most places with tables have code constructions. So click the entry, and they describe the construction (unfortunately in error correction code terms), but that can be reversed into English and actual questions to get your answer.
This technique would let you answer any problem along the line of "How many questions, each with b options, would it take to determine between N items, if there are at most d lies told?"
This type of solution requires choices and branching. Mine was nicer I think since it requires no branches, and the weighings themselves spit out the bad coin number in base 3.
It's:
X. AAAA vs BBBB
If X was balanced, XX: C1 vs C2, then XXX: C2 vs C3 (general way to distinguish 2 unknowns: weigh one unknown against one known)
If X was unbalanced, WLOG assume A was heavier. XY: A1 A2 B1 vs A3 A4 B2
Then if XY balanced, use the general method to distinguish B3 or B4 as answer.
If XY unbalanced, WLOG assume A1 or A2 is heavy, or B2 is light, so XYY A1 vs A2 finds the heavy one or implies B2 is light.
What is XY? What's WLOG?
Can you dumb this down a bit more please?
Number the balls 7 8 9 10 11 12 14 15 16 17 18 19
Then do three weighings according to rows like this:
0 1 -1 0 1 -1 1 -1 0 1 -1 0
1 1 -1 -1 -1 0 0 1 1 1 -1 -1
-1 -1 0 0 0 0 0 0 0 0 1 1
If left is heavy, record 0, else if balanced, record 1, else if right heavy, record 2.
Then the base 3 value gives the coin label above (in absolute value) and is >0 if heavier, else is -1 if lighter.
Also note there is no if/then. You simply follow these three and record the numbers which spit out the answer.
The idea is the matrix represents an error correction code in base 3. You need each column distinct, and coin j being out of whack selects column j from the matrix.
For example, say ball 7 is heavier. From top to bottom, the weightings would result in: 120. However, if ball 19 was lighter you would obtain the same weighing sequence 120.
I believe that there is another condition that must be imposed on the matrix - namely, the column must be unique, but also -1*column must not be another column.
I created the following matrix, which can identify the ball and if it is heavier or lighter. However, this does not line up neatly, and would require a table to decode the output of the weighings.
-1 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 0 1 0 -1 -1 0 1 1 0 1 0 1 -1 0 -1 1 -1 0 1 0 1 -1
For example, the sequence 001 means the first ball is heavier. The sequence 221 means the first ball is lighter. Etc.
The idea works. When I solved it before I cider it up to test
A. There are many ways to identify the misweighted ball in three deterministic weighings. The simple approach is to label the possible states 1 to 24, or alpha to omega, or what have you, and then to assign an outcome code to each state.
SideQuark wants to entwine the outcome codes with the state labels, so that the label for each state is the value represented by its outcome code when that code is interpreted as a trinary number. This sets up feedback between things that would otherwise be independent of each other; it's sort of like writing a quine.
B. Therefore, it's possible that the odd choice to start numbering the balls at 7 (and skip 13) represents a compromise with what outcome codes our system can generate.
(For example, you can identify and diagnose the misweighted ball in these three deterministic weighings: 1,2,3,4 vs 5,6,7,8; 1,5,9,10 vs 2,6,7,11; 2,4,6,9 vs 3,7,10,12. If an outcome of 0 represents the right side of the scale being heavier, and 1 represents the left side being heavier, then the outcome code 121 uniquely identifies the 4 ball as being too heavy. But 121 interpreted as a trinary number is 16, not 4.
In fact, in this weighing scheme, the outcome code "4" (= 011) cannot occur, and 9 and 26 are also impossible.)
C. SideQuark appears to be cheating in that his balls are labeled from 7 to 19 (skipping 13), but his column indices range from 1 to 12. This means he's given up on getting a real correspondence between label and outcome code (the balls have different labels depending on whether you're reading the specification of the test [1-12 in full] or the interpretation of the result [7-19 with a hole]), but it looks like he is still aiming to have a meaningful correspondence between the code that specifies a particular ball is heavy and the code that specifies the same ball is light.
D. We're going to record a trinary number between 000 and 222, a range from 0 to +26. It is not clear to me how we would get a negative number. We could interpret the number we get as a three-trit three's complement value, but in that case we'd really want the balls to be labeled from 1 to 12, so that a single value uniquely identifies and diagnoses a ball - for an example of the problem we're about to run into, the outcome 101 represents the value 10, and should tell us that the 10 ball is too heavy, but if we're using three's complement then 101 is also the value -17, which tells us that the 17 ball is too light. This is a bad outcome when the 10 ball and the 17 ball both exist.
Use base 3 with 'digits' -1,0,1.
It's useful and interesting to note you can uniquely encode and decode any integer in any base with 'shifted' digits. If you allow some digits to be positive and some negative, you get both positive and negative integers encoded for free without needing sign extraction, using standard algorithms. You can also change base per slot and go into mixed-radix tricks for encoding (useful for turning things with varying choices per slot into unique encodings).
Things look like
Encode(int val, int base, int shift)
while (val != 0)
d = ((val%b)+b)%b; // get positive mod 'digit'
if (d + shift >= b) d = d-b; // shift digits
val -= d; // kill off low digit - often not done in positive only case
val /= d;
output(d)
Decode (list of digits, int base, int shift)
val = 0
reverse digits
foreach digit d
val = val * b + d
return val
digits -1,0,1 also makes a unique base 3 representation of any integer using the same encoding and decoding algorithms. In fact, you can shift digits in any base like this and still get unique encoding and decoding, with the benefit that they handle positive and negative numbers for free.
Shifting the digits is exactly equivalent to biasing the number; in this case, changing the interpretation of the graphical symbols "0", "1" and "2" to be the numeric values -1, 0, and 1 just means we subtract trinary 111 = decimal 13 from our three-digit value. So we could represent the value -4 as "0--" (in "shifted trinary") or we could represent it as "100" (in biased trinary, since 9-13 = -4) and those representations are glyph-to-glyph isomorphic. Since the representable range of three digits of ordinary trinary is 0-26, the representable range using three of these shifted digits is -13 to +13.
Using this -/0/+ representation, it is easy to state a weighing schedule in which the absolute value of the three trinary digits you get as output from the scale represent equals the label (1-12) of the misweighted ball.[1]
But it appears to be impossible for the representation to be positive whenever the ball is heavy and negative whenever the ball is light.
Let's assume that we record a - when the scale tips right and a + when the scale tips left.
The problem is that every time we do a weighing, there will be at least one ball on each side of the scale. (Or else the weighing would generate zero information.) So without loss of generality, there is a ball L on the left of the weighing which gives us our most significant digit, and a different ball R on the right of the same weighing.
Since we record a - when the scale tips right, our most significant digit will be -, and therefore our overall result will be negative, whenever ball R is heavy. This conflicts with your stated goal of having the result be positive whenever any ball is heavy.
This will make it difficult to design an algorithm to diagnose the ball that doesn't involve any branches - we can "branchlessly" (to the extent taking an absolute value doesn't branch) determine which ball is misweighted, but we need a branch (or a lookup table) to determine whether it's light or heavy.
[1] This will do the job:
6 8 10 11 vs 5 7 9 12
3 5 7 11 vs 2 4 6 12
1 2 5 10 vs 4 7 8 11
By choosing 7-19, omit 13, you finding the center values of the range 0-26. This is important because when you write the values in ternary each column has the same number of 0 and 2 digits.
This would not hold if you simply used the values 0-12, for example. Then the first column has eight 0’s and four 1’s, so it does not correspond to a valid weighting.
It’s important to note that this method doesn’t work for the problem described - it only works if you know that the odd ball out is heavier. You can see my other comment for a description of why, and a working solution.
Here I picked the above since it is a valid sequence, but fails for the reason elsewhere on this page - you also need to not have negative paired columns. And that too turns out to not be enough, so write a simple program and crunch to check each case that does work, and there's (I think) 304. I should stop poking at this as I have real work I should be doing :)
If you really want to clean it up, you could then map the answer 3-vector of scrambled -1,0,1 values into another decoder matrix to map into values you like. Or use a look up table.
The solution is: Take one coin from the first bag, two coins from the second bag, three coins from the third bag, etc. Weigh all of the selected coins at the same time to get a total weight. Subtract from the total weight the triangular number of the number of bags[1], in ounces. The remainder, divided by 0.1, will be the number of the bag with the fake coins.
[1] eg, for three bags, 1 + 2 + 3 = 6
One of the Die Hard movies had the "extract 1 gallon from a 3 gallon and a 5 gallon jug" puzzle in it.
The only requirement is to have a unique number of coins from the fake coin pouch. Since we don't know which one that is, every pouch gets a different number of coins, and not necessarily a sequence starting from 1. Then, you weigh them all assuming they're all real, but wait, they're not! So you take the excess weight, divide it by the excess weight per fake coin, and the bag from which you took out that many coins is the bag with the fake coins.
Weigh 1: 9 vs 9 (18 on balance), 9 off
Weigh 2: 3 vs 3 (6 on balance ), 3 off
Weigh 3: 1 vs 1 (2 on balance ), 1 off
....and now I understand, you would only initially know there is a difference in weight, not which side had the heavier or lighter ball.
18 balls have 36>27 states.
This is a very relatable feeling. I often find myself getting trapped into thinking about problems I am unlikely to solve.
There's a portion of my brain that's immediately hijacked and starts thinking about the problem whether I'm actively working on it or not.
Happens to me for work problems outside of work, too. I'll just be chilling in the shower or something and suddenly I'm thinking about a work problem.
Maybe one day I'll retire from programming and do something less theoretical e.g. driving a semi.
I have this problem as well. The most disturbing thing is retracing old paths to dead ends without realizing it. Here I mean internal mental models, but also applies to previously known results I just wasn't familiar with, and don't turn up unless specific phrases are known.
Abstract:
We show that for any union-closed family F\subseeq 2^[n],F≠{∅}, there exists an i∈[n] which is contained in a 0.01 fraction of the sets in F. This is the first known constant lower bound, and improves upon the Ω(log_2(|F|)^{−1}) bounds of Knill and Wójick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if A,B are independent samples from a distribution over subsets of [n] such that Pr[i∈A]<0.01 for all i and H(A)>0, then H(A∪B)>H(A).
Impressive result however.
It is possible for someone who is not a full-time mathematician by still working on mathematics in their spare time or in a mathy-field in their day job but very very few real examples of that can be found.
So I don't find the title clickbaity. It is really really hard to come up with results like this without working on mathematics research (not AI research) 10 hours a day!
Always enjoy seeing this sentiment reaffirmed. Funny that Gilmer states it in a somewhat self surprised tone.
Rings of the Einstein quote:
“[I do not] carry such information in my mind since it is readily available in books. He also said, “…The value of a … education is not the learning of many facts but the training of the mind to think.”
Especially in poorer public schools, learning is modeled as recitation. It's awfully sad. Most students are lucky if they get one or two teachers that break that mold.
While anyone with a (good) STEM education knows this to be obviously true, its frightening how many people go through 16+ years of formal schooling where they mostly do rote memorization and regurgitation.
That's how most education is structure, though. Unless your point is simply that, though it reads as if you're blaming the student.
We measure students with tests, but all we get are people good at taking tests and rote memorization.
Maybe if we had a better way to access actual learning but are teachers even paid enough to truly care?
In many other disciplines applying what students learned to things they’ve never seen before isn’t really possible.
For example, you can teach students what factors were needed to start the Industrial Revolution and why, but then, to answer “why didn’t the Industrial Revolution start in the 1750s in South America?” students would have to know about South America in the 1750s. If they don’t, all they can answer is “it must be because not all of the prerequisites foo, bar, baz, … were available, but I wouldn’t know which one(s)”.
Because of that, that question isn’t better than the dull “why did the Industrial Revolution start where and when it started?”, the answer to which can be learned by rote learning.
There's no incentive to pay teachers more; system is working as designed. Those students aren't giving short-term quarterly gains.
Of course any non-STEM field has geniuses, but our impressions—like the parent’s—are formed from averages, and STEM majors are smarter on avg.
I think it's the first place I ever saw the "Information Diagram" (https://en.wikipedia.org/wiki/Information_diagram) which is a Venn-like diagram that really helps to remember all these relationships.
It depends...good luck doing math if you don't have a good grasp of arithmetic . Having to look up everything would be a major chore.
> “I was a bit rusty, and to be honest, I was stuck,” Gilmer said. “But I was eager to see where the community would take it.”
> Yet Gilmer thinks the same circumstances that left him out of practice probably made his proof possible in the first place.
This reminds me of the principle of Shoshin: https://en.wikipedia.org/wiki/Shoshin
The idea that the mindset of a beginner can sometimes find insight where the mind of an expert cannot, and that experts should cultivate thinking like a beginner in some instances.
Edit: From the paper, the family F is defined as "F ⊆ (subset or equal) 2^n". I think that means for n elements there are at most 2^n subsets? In other words, no duplicated subsets. Which still doesn't clear up my confusion--5 subsets is < 2^4.
Edit: Duh, thanks for correcting my thinko.
"A collection, or family, of sets is considered “union-closed” if the union of any two sets in the family equals any existing set in the family."
EDIT: In your hypothetical collection of sets, the union of "2" and "3" would produce the set "2,3" which is not an enumerated set in your list.
Should the title of this HN entry be updated accordingly?
As for using AIs as mathematical proof helpers, I threw it the following softball and can't say that mathematicians should be too worried just yet:
>> Please prove that the first decimal digit of pi is 3.
> The value of pi is an irrational number, which means that it has an infinite number of decimal digits that do not repeat in a pattern. It is not possible to prove that any particular decimal digit is the first decimal digit of pi, because the decimal representation of pi is infinite and does not repeat. However, the first few decimal digits of pi are widely accepted to be 3.14, so the first decimal digit of pi is generally assumed to be 3.
It is not clickbait just because you interpreted it wrongly. The AI in the title only aims to demonstrate his state of out-group in regards to math.
He's more "in" that problem space than a professional differential topologist.