Solve this if you are a genius

Solve this if you are a genius

This “solve this” quiz showed up on my Facebook news feed.

8 = 56
7 = 42
6 = 30
5 = 20
3 = ?

The pattern looks pretty easy, right?

8 × 7 = 56
7 × 6 = 42
6 × 5 = 30
5 × 4 = 20

So what is the correct answer?

Here’s mine:

3 = 3 — even if it’s preceded by a bunch of false statements.

Do you agree?

Update. This is an old post and I thought it had seen all the life it was going to see. Then Angel comes and leaves a comment:

It turns out that the ‘?’ can be whatever we want it to be!

Angel backs up his claim with math. I think we have the definitive answer.

<3

Author: Brent Logan

Engineer. Lawyer. WordPress geek. Longboarder. Blood donor. Photographer. More about Brent.

104 thoughts on “Solve this if you are a genius”

      1. 3 never could be 6, becoz if u got the answer pattern accordingly to the statement given, 6 is already = 30, so 3 cannot be 6 anymore.

      2. 3 = 3 only in base 10.
        3 does not necessarily equal 10 in other bases.
        For example 3 = 11 in base 2.

        So I was thinking the problem could be solved by induction??

    1. Alba, yeah, I see the pattern. What grates on me is how they misuse the equals sign. The first four lines make four false statements, but in a pattern that can be inferred. If one is willing to make one more false statement in the same pattern, then you are correct (yet wrong…) to say 3 = 6.

      I rebelled against making a false statement, and gave the literally correct statement of 3 = 3.

      1. The only problem with this argument is, that the 4 = 12 isn’t there, that is just something that is assumed. The only pattern you can see, is that the difference between the numbers after = is, that the difference to the numbers above is equal to 2x the numbers before the =. Which means that the correct answer is 3 = 14!

  1. The answer is 9. The multiplier is the trick, nothing more. Each multiplier is one more than the one before it: 3 x 3, 5 x 4, 6 x 5, 7 x 6, etc.

  2. The answer could also be 3 = 4 based on the following pattern:

    8 = 56 (sum of all 3 digits is 19)
    7 = 42 (sum … 13)
    6 = 30 (sum … 9)
    5 = 20 (sum … 7)
    3 = 4 (sum … 7)

    The difference between the sum of the 3 digits decreases by 2 each time: 6, 4, 2, 0. It’s a ‘zero sum’ pattern.

  3. Brent, although I agree with your comment that 3 = 3 despite a number of prior false statements, my inclination (purely on aesthetic grounds) is to state that:

    3 = (insert any number but 3)

    Thus we end up with a list of false assertions, as opposed to some false followed by one true.

    1. Simon, I admit there is a symmetry to continuing the false statements. I’m not convinced, “That’s the way we’ve always done it” is a good enough excuse to knowingly be wrong, though. There is no improvement without change… ;-)

  4. Haha, thanks for this post. It just came through my LinkedIn today and it was driving me nuts.

    I feel vindicated that someone felt the same as me 3 = 3!

  5. Great, I am not alone. My answer is 3.

    For sure 3 equals 3; 3 = 3.

    The statements before are all logically wrong – they just there to change the readers mind. They would be correct if the equal symbol would be replaced by congruence relations (symbol by THREE lines) and the congruent modulo would be a part of a mathematical sequence (sentential logic symbol required). But even the sequence would be incomplete (missing 4)- so there are misleading statements, and most of the persons do interpret a sequence and congruence relation – and they confirm their wrong result as correct on the basis of previous wrong given results. IF someone thinks that 3 = 9, I would like to give him/her 3 Euros and I want to have 9 Euros back – it would be fair, because it is equal!

  6. Markus, I had mentioned 9 as the answer on LinkedIn, as everyone were answering 6… Anyways can you and Brent pls explain as how is the Answer 3 ….and yes I do like brain storming.

    Cheers Gautam.

    1. Gautum, I think Markus does a pretty good job of explaining our position: the equals sign has meaning. To be true to its meaning, the answer mus be “3 equals 3.” All other answers ignore the mathematical meaning of the equals sign and treats it like a generic symbol dividing two columns of numbers.

  7. I do not like the 3 = 14 pattern. It does not make mathematical sense. It sounds good until you attempt to prove it mathematically…
    8 = 56
    7 = 42 (56 – 42 = 14 = 2 × 7)
    6 = 30 (42 – 30 = 12 = 2 × 6)
    5 = 20 (30 – 20 = 10 = 2 × 5)
    3 = 14 (but 20 – 14 = 6, not 14 so the pattern becomes unstable, lending support to 3 = 6).

    3 = 4 in a zero sum is interesting, though, as is 3 = 9 (though it assumes the 4 is not missing which, I suppose, is fair), but 3 = 6 is by far my favorite…just because of the pure mathematical pattern. In a puzzle situation, it is safe to assume a portion of the formulae is missing. That is what makes it a puzzle.

    Those who think 3=3 just have no sense of the beauty and romance in numbers, lol. You are far too pragmatic.

  8. I saw this today and immediately I was tempted to say 3 = 6. But looking at it closely I realised 4 = 12 is not among the sequence and hence we cannot prove 3 = 6 unless we make the assumption that 4 = 12 which would be wrong.

    My answer is 3 = 12.

    Luetkemeyer made some good analysis but I think 3 = 12 would be better suited with that analysis.

    8 = 56
    7 = 42 (56 – 42 = 14 = 2 × 7)
    6 = 30 (42 – 30 = 12 = 2 × 6)
    5 = 20 (30 – 20 = 10 = 2 × 5)
    3 = 12 (20 – 12 = 8 = 2 × 4)

    Looking at it you see that the difference between the numbers on the right are reducing by 2.

      1. Like you said 5 does not equal 20 so is 6 not equal to 30. The numbers on the left and the equality sign are just mere distractions and do not relate with the numbers on the right.

  9. I see this pattern. What’s wrong in my thinking? I am not adding another series in it. Number is squared and then subtracted by the same number. Is this correct?

    82 – 8 = 56
    72 – 7 = 42
    62 – 6 = 30
    52 – 5 = 20
    32 – 3 = 6

    1. Ding! Ding! Ding! I think we have a winner.

      I love this pattern.

      Interestingly, n2 – n = (n × n) – n = n × (n – 1), which is the pattern I mentioned in my original post (minus the snarky final line…).

      1. Because of this i also thought of this:

        ALL NATURAL NUMBERS ARE EQUAL.

        The proof starts with a formula that am = an ⇒ m = n

        E.g.:

        Let a = 1 then … = (1)-3 = (1)-2 = (1)-1 = (1)0 = (1)1 = (1)2 = (1)3 = … = 1. Then … = -3 = -2 = -1 = 0 = 1 = 2 = 3 = …

        Let a = -1 then … = (-1)-3 = (-1)-2 = (-1)-1 = (-1)0 = (-1)1 = (-1)2 = (-1)3 = … = 1. Then … = -3 = -2 = -1 = 0 = 1 = 2 = 3 = …

        Let a = 0 then … = (0)-3 = (0)-2 = (0)-1 = (0)0 = (0)1 = (0)2 = (0)3 = … = 1. Then … = -3 = -2 = -1 = 0 = 1 = 2 = 3 = …

        So always remember am = an ⇒ m = n but a ≠ -1, 0, 1

      2. Zoheb, hmmm… I’m not sure why you don’t like my algebra. Do you not agree that n2 – n = n(n – 1)? Or, to use an example, 82 – 8 = 8 × 7.

        By the way, it might not be obvious to others that you are attempting (maybe?) to link to a clearer representation of your proof here.

  10. It’s different PROBLEM that has been addressed in that blog. I just wanted to throw that in your post so I just posted it here.

  11. Interesting mental exercise. The facts are: equations in math (and chemistry) MUST be balanced otherwise our world would end.

    First: Math rules state that if you perform an operation to one side of the equation you must do the same to the other side.

    Second: There is no rule at the beginning of this puzzle that sets up a pattern — it doesn’t say 8 × X = 56 so assuming that is incorrect. None of the implied patterns work either because the moment you introduce an operation you are changing the equation completely.

    As Brent states, all of the equations up to 3 = ? are false. 8 does NOT equal 56.

    There’s no proof needed because we know that to be true.

    In the scientific world many scientist perform experiments and come to answers that are later disproven. Just because someone states that 8 = 56 doesn’t make that true. In fact it’s quite interesting to note that most people’s answers do not follow any mathematical method. Most are happy to state 3 = 6. Faced with a series of incorrect data most people would continue to interpret that as true and solve a math question incorrectly.

    That’s a little troubling.

    1. So, Robert … let’s see if we can recover this. Instead of the equals sign, let’s put a meaningless divider in it’s place. Then include the instructions, “Continue this pattern.”

      What would you put in place of the question mark? ;-)

      1. But that’s not the puzzle.

        I get what you’re saying but then I could easily say “put a × (times) sign at the beginning preceded by variable Y and then solve.”

        I must adhere to the strictest rules of mathematics. If not we are altering the results.

        But to answer your question and adding elements to the equation to make it “work” then the answers would be

        7 × 8 = 56
        6 × 7 = 42
        5 × 6 = 30
        4 × 5 = 20
        3 × 4 = 12 (missing)
        2 × 3 = 6.

        But I state again that interpreting the original data and ADDING data to find the solution is incorrect

    1. Ha! So they gave the answer to start with? :-D

      This brings to mind the saying, “The first step to solving a math equation is admitting you have a problem.”

      Thanks for dropping by. :-)

  12. 3=13

    8 = 56; 56 – 8 – 7 + 1 = 42 (next result)
    7 = 42; 42 – 7 – 6 + 1 = 30
    6 = 30; 30 – 6 – 5 + 1 = 20
    5 = 20; 20 – 5 – 3 + 1 = 13
    3 = 13

    1. Chiunti, although your answer looks different from everything before it, I don’t believe it is, plus it doesn’t explain the first pair of numbers. Here’s why I say that.

      Your equation is effectively:

      An = An+1 – (n + 1) – n + 1

      In other words, the answer for 7 is the answer for 8 less 8 less 7 plus 1. What happens if we say the answer for n + 1 is (n + 1) × n, substitute it into your proposal, and then simplify? I suggest doing this because it shouldn’t detract from your answer, and it allows us to explain the first number pair.

      An = (n + 1)n – (n + 1) – n + 1
      An = (n + 1)n – n – 1 – n + 1
      An = (n + 1)n – n – n
      An = (n + 1)n – 2n
      An = (n + 1 – 2)n
      An = (n – 1)n
      An = n(n – 1)

      What do you think?

      1. Considering:
        N1 = A1

        Nn = An

        then:

        An = A(n-1) – N(n-1) – Nn + 1

        N will never have value of 4, because it is not in the numbers listed.

        I hope I explained.

    2. Chiunti. There is an error in your interpretation. You forgot the “4” line which is obviously missing from the pattern

      8 = 56; 56 – 8 – 7 + 1 = 42 (next result)
      7 = 42; 42 – 7 – 6 + 1 = 30
      6 = 30; 30 – 6 – 5 + 1 = 20
      5 = 20; 20 – 5 – 4 + 1 = 12
      4 = 12; 12 – 4 – 3 + 1 = 6
      3 = 6; 6 – 3 – 2 + 1 = 2

      Therefore in your pattern 3 = 6 is next in the sequence preceded by 4 = 12

      Following 3 you have

      2 = 2; 2 – 2 – 1 + 1 = 0

  13. let suppose 8 = a
    7 = b
    6 = c
    5 = d
    and 3 = e
    then 3 = 12 as the series goes on, but this assumption is proved logically and there will be no questions as how 8 equals 56 so. i guess my answer is interesting.

  14. if we dont assign numbers then we take them(left side numbers) as symbols only so then 4 will not be a concern then this question can be solved logically and 3 will be 12 and no issues will be their as how 8 equals 56. 7 equals 42 and so on

    1. Those “thingies” on the left side of the equals signs already are symbols. Unlike the letters you’re putting on the right side, numbers have a fixed value. Once you equate the letters to the numbers, you’re assigning fixed values to them as well (at least for this context). I’m not convinced you’re solving my original (and somewhat facetious) issue.

      I’m more interested what logic you’re using to determine that the question mark should be replaced with 12.

    1. I see your pattern, but it seems to ignore that the last step for numbers on the left decreased by two. It doesn’t seem that it matters for your pattern (and maybe it shouldn’t…).

      Thanks for dropping by!

  15. Wrong.. These things intentionally omit numbers. To assume “what if” is not Logical.. There is always a MOST logical conclusion without hypothetical added numbers.. The correct answer is 12

    Answer … This formula stays consistent using only the Logic provided without assuming anything. The left numbers mean absolutely nothing in finding the answer other than validating the Proven Facts… SO, using ONLY the right side numbers follow this formula the validate with the left corresponding number >>> 56 – 42 = “14” then 42 – 30 = “12” then 30 – 20 = “10” then ?? from 20 will get the next in sequence figure? Note the factors reduce in perfect order will give to you the next in sequence figure of “8”? Note 14 / 12 / 10 / 8 <>> 3 x 4 = What??? Bingo TWELVE (12) … 12 is the only logical answer.

    1. If you will only see, the Left corresponding number multiplied by the next Left side number gives to you the right side number . You will see the 4 was omitted. However in the last row, the $ is called in as the next adjacent multiplication factor… Thus 3 x “4” = 12. The absolute most logical conclusion can only be 12 as the answer

  16. Look the results :

    Results for the left-hand numbers: 5, 4, 3, 2, ?; ? and the right numbers for 6, 2, 0, 0,?,?

    To see if the numbers are based on logic; the size of the result is 6, because

    8 × 7 = 5 6
    7 × 6 = 4 2
    6 × 5 = 3 0
    5 × 4 = 2 0
    4 × 3 = 1 2
    3 × 2 = 6
    2 × 1 = 2
    1 × 1 = 1

    results number – left rows numbers : 5, 4, 3, 2, 1; right: 6, 2, 0, 0, 2, 6

    So the result is number 6

    :)

  17. Simplest is: x = x*x – x

    When you assume ‘what if 8 = 56’ etc.

    The answer is 6 also when you visualize numbers so that the number on right is divided by the number under it on left, e.g. 56/7.

    So, you get this:
    8 = 56/7
    7 = 42/6
    6 = 30/5
    5 = 20/4
    4 = a?/3
    3 = b?/2

    a = 12
    b = 6

    And proof is: x = x*(x-1)/(x-1)

    E.g. 6 = 30/5 or  6 = 6*5/5 

    Meaning x = x 

    ;-)

    And if you want to be strict with the puzzle then the most correct answer is that…

    3 = any number but 3

    …because all statements are wrong, so to make your right you have to keep it wrong so all answers are in ‘harmony’ (same ‘logic’).

    But if I have to pick a wrong number then it is 6, because it is beautifully wrong to state 3 = 6 in context of this interesting thought-provoking puzzle.

      1. Hi Brent! Was a pleasure to find your blog and see your answer and that of others. Just shows how differently different people approach the same thing :) Technically speaking yours is most correct…

    1. Dimas, interesting pattern. I like how it doesn’t worry about there not being a row starting with the number four. Unfortunately, it doesn’t explain where the 56 comes from, while the n × (n – 1) pattern does.

      Thanks for dropping by and writing a comment. :-)

  18. I lost what I mistakenly thought was a friend over this puzzle, she tried to bully me into saying the answer was 6. That may be the most popular but not necessarily the right answer, not the way I see it.

    I see it as 0, used to think it was 12.

    8 = 56
    7 = 42
    6 = 30
    5 = 20
    4 = 12
    3 = 6
    2 = 2
    1 = 1
    0 = 0 with the answer being 0.

  19. Math makes my head hurt; but I love how you guys have an obvious love for it. I wish I understood it better.

    1. jess2248, doing a quick run through the names, I’d guess we’re not all guys. There are some gals in the group. :-)

      And yes, math is fun. Math is a language and a tool to understand how the world works, in so many ways. And even though I like math, it can still make my head hurt. ;-)

  20. I am not mathematical, but I am logical. What I got when I looked at this was 15. To me the pattern went like this:

    15

    8 = 56; (56/8 = 7)
    7 = 42; (42/7 = 6)
    6 = 30; (30/6 = 5)
    5 = 20; (20/5 = 4)
    3 = 15; (15/3 = 3)

      1. normdeploom (love that name!), I think this is the same pattern many have posted, but you’ve rearranged it slightly. Instead of saying 8 × 7 = 56 you say 56/8 = 7.

        What is different is how you interpret this pattern. For you, the important point seems to be that the number on the right decreases by one for each row. Thus the final row should be 9/3 = 3, with ? = 3 being your final (corrected) answer. Right?

        If so, you got the same answer as Jon, and for pretty much the same reason.

        Thanks for dropping by!

  21. Finally !!!!! I’ve been answering this question one hundred times, and it’s the first time I see someone who’s finally smart enough…. Yes, 3 = 3 …. other statements are false !!!!!

    God, it took me more than a year to find you !!!! lol

    1. Yet another pattern. I’m not thrilled how each row is not self-contained, but instead relies on the next. That makes the 8 in the first row maybe arbitrary, and definitely useless. It does sidestep (or maybe just ignore) the skipped 4. That might be a strength instead of a fault.

      Thanks for dropping by!

  22. It turns out that the ‘?’ can be whatever we want it to be!

    Allow me to illustrate. Take for example,

    f(x) = (1/40)x4 – (13/20)x3 + (291/40)x2 – (553/20)x + 42

    and evaluate f(8), f(7), f(6), f(5), and f(3). It turns out that:

    f(8) = 56,
    f(7) = 42,
    f(6) = 30,
    f(5) = 20,
    f(3) = 9.

    Wait, what sorcery is this? Turns out that although the popular rule f(x) = x(x – 1), which gives f(x) = 6, satisfies the known values in the sequence, that

    f(x) = (1/40)x4 – (13/20)x3 + (291/40)x2 – (553/20)x + 42

    also satisfies them–except with a different value of f(3)!

    Here’s another one that also works but gives f(3) = 12:

    f(x) = (1/20)x4 – (13/10)x3 + (271/20)x2 – (543/10)x + 84

    And here is one where f(3) = ?

    f(x) = (1/120)(? – 6)x4 – (13/60)(? – 6)x3 + (1/120)(251? – 1386)x2 + (1/60)(3138 – 533?)x + 14(? – 6)

    Finally, in general if you want the ‘?’ = k, i.e., f(3) = k where k is the value of your choice, then

    f(x) = (1/120)(k – 6)x4 – (13/60)(k – 6)x3 + (1/120)(251k – 1386)x2 + (1/60)(3138 – 533k)x + 14(k – 6)

    More details here.

    1. Angel, wow! That is really cool. And to think I had thought the discussion on this problem had been completely played out.

      I love how for the k = 6 case, the function simplifies to f(n) = n(n-1).

      And btw, I appreciate your using the relation operator in your linked paper. ;-)

      Thanks for dropping by and commenting. :-)

      1. More than seven years later, this problem continues to intrigue people. I’m glad you found it interesting and, ignoring my problem with the equals sign, I agree with you.

        Thanks for the visit and comment!

  23. Brent Logan! You truly do have a great mind. I was almost convinced that 3 does not equal 3! My 6 year old dog was right all along. Thanks for enlightening me.

    The amusement continues….

  24. Well I didn’t use anyone else’s logic, but I came to the conclusion that the answer is 3. Not even bringing 4 × 3 into the equation but my reasoning is 3 could only equal itself or 0. I’m not great at maths but that’s my answer.

  25. gtt, you had me until you said that 3 could equal 0. But then you recovered by saying you’re not great at maths, so I think (once again) we agree. ;-)

    Thanks for dropping by. :-)

  26. 8 = 56
    7 = 42; 56 – 42 = 14; 14/2 = 7
    6 = 30; 42 – 30 = 12; 12/2 = 6
    5 = 20; 30 – 20 = 10; 10/2 = 5
    3 = 14; 20 – 14 = 6; 6/2 = 3

  27. Nathan, you’re not the first to propose 14, but I think you are the first to use this reasoning. I’ll have to go back and review and see whether or not yours is mathematically equivalent to another’s.

    Does it bother you that 4 is skipped? It seems like you could generate the missing row for 4 and then would end up with a different answer for 3.

  28. I may also elaborate on the “?” Being whatever we want it to be claim. Originally I figured the solution to be 6, holding no bearing on the equivalence statements. First of all, we are assuming that these numbers lie within the set of Real Numbers in order to apply various properties. This is valid since the integers do indeed lie within. I do want to point out that considering mathematics as a language and taking a crack at this problem it must retain logic. So, by saying that 3=3 is the answer is indeed false under the presumption that this so-called equivalence relation is false which means that the last statement must retain a false statement, so it could be any number not equal to three. There is really no reason to consider otherwise as it would conjure up logic that really isn’t there. In order to consider a sequence, it has to be a function and still abides by the equivalence statement. So, saying it can be anything is indeed illogical. However, assuming there is an underlying approach to be considered, we would have to define what “=” really is as a relation in this case. Consider the problem given:

    8 = 56
    7 = 42
    6 = 30
    5 = 20
    3 = ?

    We know that anything divided by 1 is itself due to the multiplicative identity axiom which means that the remainder of this quotient would be zero.

    So, I can arbitrarily define the relation to be “=mod(1)”

    So that 0=0 for each case.

    Hence, ? Would indeed be any number. One could check to see if any other relation would work, but this one is guaranteed. However, like I said before this method does not retain the logic that should be taken into consideration when taking an unbiased glance at the problem.

    1. You are correct in that every number can only equal itself. However, logic should flow as follows:

      “If not B ⇒ not A” then
      “A ⇒ B” and this type of logic can work for “if and only if statements”.

      So, if we are considering the statements given it can be read as follows:

      “If false and false and false and false” then what is the last statement?”

      It must be false.

      Because, the negated statement should be a true implying the rest being true. This is only logical when “3 = anything but three.”

      That way when you negate that statement you have:

      “If true, then true and true and true and true”
      So that “if 3 = 3, then 5 = 5, 6 = 6, 7 = 7, and 8 = 8”

  29. This is not a math problem. It is a purely a question of logic. Establish the basic pattern working from bottom up. Double the number on the left side of the equal sign and add it to the number in the right side, the sum of which equals the number above it. Repeat working upwards. Works going down too but subtracting obviously. And the creators left the 4 out for a reason, you can’t just add it in inorder to make your pattern work. ? = 14.

  30. However, your solution does not utilize logic. Also, math is built off of logic, so they go hand-in-hand. You are assuming that there is a pattern which assumes the false equalities are indeed true which is contradictory. This is what I was attempting to explain previously. The most logical approach is to assume that in order to obtain a string of true statements one has to start with the false statement since that is what is implying the following statements.

    It is also a simplification to state that the 4 is left out for a reason. Take for instance the sequence give, as previously mentioned.

    f(n) = n(n-1) ⇒ f(8) = 56, f(7) = 42, f(6) = 30, f(5) = 20, f(4) = 12, f(3) = 6.

    So, if one wants to illogically create a false equality between n and n(n-1) then you have a valid pattern. In general, most patterns do not include every term up to the term that is desired. Take for instance a series problem stating that the first term of the series is, say, 1 and that each consecutive term is increase by 8. Let n be contained within the natural numbers union with the singleton set 0. This allows n = 0, 1, 2, …

    Now, find the 25th term of the series.

    Writing out this series we have that the nth term can be found as:

    A0 = A0, A1 = A0 + 8, A2 = A1 + 8 = A0 + 2(8), A3 = A2+8 = A0 + 2(8) + 8 = A0 + 3(8), ……. An = A0 +n(8)

    The first term is A0 = 1 which is given for n = 0. Hence, the 25th term is determined for n = 24. This implies that A24 = 1 + 24(8).

    So, we could still determine this pattern if we were given every term up to n – 1, because we have generalized the series. We are also able to find every term in between some kth term and the nth term. This is how patterns work.

  31. Yea, pretty sure it comes down to intent, or what the intent of the creator was originally.

    8 = 56
    7 = 42
    6 = 30
    5 = 20
    3 =

    We can assume the author wanted us to multiply starting with 3 similar to the following:

    8 × 7 = 56
    7 × 6 = 42
    6 × 5 = 30
    5 × 4 = 20
    3 × 3 = 9

    But I do not believe that would be intuitive or logical to assume that would and could be the only right answer. Please let me explain.

    8 × 7 = 56
    7 × 6 = 42
    6 × 5 = 30
    5 × 4 = 20
    4 × 3 = 12
    3 × 2 = 6

    As you can see, the pattern remains the same. 3 × 4 = 12; 4 × 5 = 20; 5 × 6 = 30, and so on. I say it comes down to the intent of the author.

    1. I agree that there can be various interpretations, but I was trying to establish the irrefutably most logical solution. Irrespective of the intent of the author. The reason being that conjuring up any particular pattern is in contradiction to the logic set in place.

    1. Haha! I was going to tell you to read through all of the comments to find the answer, but I reread the post first. I think you’ll find my answers (both of them) near the end.

      Thanks for dropping by and commenting. :-)

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