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How can you make the number 12 with just the numbers 1,5,19,10? You can only use each number once, and can only use subtraction addition multiplication and division.

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    $\begingroup$ Do we need to use each of 1,5,19,10 exactly once? $\endgroup$
    – Rand al'Thor
    Commented Apr 7, 2017 at 19:32
  • $\begingroup$ Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. If not, some responses to the answerers to help steer them in the right direction would be helpful. $\endgroup$
    – Rubio ♦
    Commented Apr 18, 2017 at 6:36

5 Answers 5

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It's certainly possible if

the division operation is what is sometimes termed "integer division" (that is, ignoring remainders)

Since

$19 \text{ div } 5 = 3$

So

$10 + 19 \text{ div } 5 - 1 = 10 + 3 - 1 = 12$

A brute force of non-lateral-thinking evaluations...

If for each of the $4!=24$ permutations of the numbers [A, B, C, D] we take each of the $4^3=64$ three-wise Cartesian product of the four operations (+, -, *, /) [p, q, r] and arrange them in the sixteen (I think) ways:

A
A p B
A p B q C
(A p B) q C
A p (B q C)
A p B q C r D
(A p B) q C r D
A p (B q C) r D
A p B q (C r D)
(A p B) q (C r D)
(A p B q C) r D
A p (B q C r D)
((A p B) q C) r D
A p ((B q C) r D)
(A p (B q C)) r D
A p (B q (C r D))

Then (while we will have equivalents) we will cover all the possible ways to calculate a result without resorting to any lateral thinking approach. (Note that we do not need B or C q D and the like since the permutations and Cartesian product will rearrange to equivalents in terms of A or A p B and so on.

Note: Please do comment if there are any I have missed! (I have not put the formula in terms of Catalan numbers, as I am not 100% on it myself)

This gives only $4^3\times 4!\times 16 = 24,576$ evaluations to make, which makes it pretty easy for a computer to print any that are as close or closer to $12$ as any so far evaluated - as this Python 3 code does (I added an epsilon just to catch any floating point deviations):

from itertools import permutations, product
distance = 12
epsilon = 0.001
for A,B,C,D in permutations((1,5,10,19)):
    for p,q,r in product('+-*/', repeat=3):
            for s in ("{0}", "{0}{4}{1}", "{0}{4}{1}{5}{2}", "({0}{4}{1}){5}{2}", "{0}{4}({1}{5}{2})", "{0}{4}{1}{5}{2}{6}{3}","({0}{4}{1}){5}{2}{6}{3}","{0}{4}({1}{5}{2}){6}{3}","{0}{4}{1}{5}({2}{6}{3})","({0}{4}{1}){5}({2}{6}{3})","({0}{4}{1}{5}{2}){6}{3}","{0}{4}({1}{5}{2}{6}{3})","(({0}{4}{1}){5}{2}){6}{3}","{0}{4}(({1}{5}{2}){6}{3})","({0}{4}({1}{5}{2})){6}{3}","{0}{4}({1}{5}({2}{6}{3}))"):
                    e = s.format(A,B,C,D,p,q,r)
                    v = eval(e)
                    curDistance = abs(12 - v)
                    if curDistance - epsilon <= distance:
                            distance = curDistance
                            print(e, '=', v)

Which prints:

1 = 1
1+5 = 6
1+5+10 = 16
(1+5)+10 = 16
1+(5+10) = 16
1+5+10 = 16
(1+5)+10 = 16
1+(5+10) = 16
1+5+10 = 16
(1+5)+10 = 16
1+(5+10) = 16
1+5+10 = 16
(1+5)+10 = 16
1+(5+10) = 16
1+5-10+19 = 15
(1+5)-10+19 = 15
1+(5-10)+19 = 15
(1+5-10)+19 = 15
1+(5-10+19) = 15
((1+5)-10)+19 = 15
1+((5-10)+19) = 15
(1+(5-10))+19 = 15
1+5-(10-19) = 15
(1+5)-(10-19) = 15
1+(5-(10-19)) = 15
1+5/10*19 = 10.5
(1+5)/10*19 = 11.4
((1+5)/10)*19 = 11.4
(1+5)/(10/19) = 11.4
(1+5)*19/10 = 11.4
(1+5)*(19/10) = 11.399999999999999
((1+5)*19)/10 = 11.4
(5+1)/10*19 = 11.4
((5+1)/10)*19 = 11.4
(5+1)/(10/19) = 11.4
(5+1)*19/10 = 11.4
(5+1)*(19/10) = 11.399999999999999
((5+1)*19)/10 = 11.4
19*(1+5)/10 = 11.4
19*((1+5)/10) = 11.4
(19*(1+5))/10 = 11.4
19*(5+1)/10 = 11.4
19*((5+1)/10) = 11.4
(19*(5+1))/10 = 11.4
19/10*(1+5) = 11.399999999999999
(19/10)*(1+5) = 11.399999999999999
19/(10/(1+5)) = 11.4
19/10*(5+1) = 11.399999999999999
(19/10)*(5+1) = 11.399999999999999
19/(10/(5+1)) = 11.4

and shows that the closest we can get is $11.4$ (subject to me being correct about the enumeration!)

Note to run this in Python 2, change for A,B,C,D in permutations((1,5,10,19)): to for A,B,C,D in permutations((1.,5.,10.,19.)): in order to correctly evaluate the floating point arithmetic

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    $\begingroup$ Excellent tour de brute force (and exhibition of interesting Python utilities)! I don't see anything missing but do suspect some duplication, as A p B q C, for instance, seems to be covered by (A p B) q C and A p (B q C). $\endgroup$
    – humn
    Commented Apr 8, 2017 at 16:58
  • $\begingroup$ Yes plenty of duplication - note that in your "for instance" not all (p, q) choices have all three the same, some do, while others have two equal and a third different and some are one pair, while others are another - the duplication just avoids bothering to write complicated code at the expense of more evaluations. (Think A - B / C and A / B - C and their parenthesised counterparts. EDIT: Maybe the reordering actually takes care of these, um...) $\endgroup$
    – Jonathan Allan
    Commented Apr 8, 2017 at 17:05
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Turn the 19 upside-down to get 61, subtract 1 to get 60, divide by 10 - 5 to get 12. voilà.

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  • $\begingroup$ What about the 10? (Also, I don't think your operation counts under the "subtraction addition multiplication and division" specified in the question.) $\endgroup$
    – Rand al'Thor
    Commented Apr 7, 2017 at 21:15
  • $\begingroup$ turn that upside-down too, and multiply. he doesn't say we have to use all the numbers - implicitly i guess we should $\endgroup$
    – JMP
    Commented Apr 7, 2017 at 21:17
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    $\begingroup$ @boboquack Not every short answer is necessarily a bad one. $\endgroup$
    – Rand al'Thor
    Commented Apr 7, 2017 at 22:18
  • $\begingroup$ If you're going to count an upside down 10 as 1, why not just flip that and subtract 7 from 19? $\endgroup$
    – Ian MacDonald
    Commented Apr 8, 2017 at 16:57
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Do you really mean 19? The order of those numbers looks weird, so I'm wondering if you made a typo and actually mean 1, 5, 9, 10. If so, the answer is easier:

$\frac{10}{5}+9+1=12$.

If you really do mean 19 and not 9, then it's harder to get to 12. I've come up with a few possibilities that almost work:

  • $\frac{(19+5)\times5}{10}=12$ (uses 5 twice, doesn't use 1)

  • $10+\sqrt{\frac{19+1}{5}}=12$ (uses square root)

  • $10+\frac{19+1}{10}=12$ (uses 10 twice, doesn't use 5)

... and so on, but nothing that quite makes it perfectly.

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I'll add another push-the-envelope answer.

I haven't found any solution in base 10, but there are solutions when the numbers are in other bases, for example:

   12₁₁ = 19₁₁ + 5₁₁ − 1₁₁ − 10₁₁ = 20 + 5 − 1 − 11 = 13
   12₁₂ = 1₁₂ × 5₁₂ + 19₁₂ − 10₁₂ = 1 × 5 + 21 − 12 = 14
   12₁₃ = 1₁₃ + 5₁₃ + 19₁₃ − 10₁₃ = 1 + 5 +22 − 13 = 15
   12₂₈ = 19₂₈ − 10₂₈ / (5₂₈ − 1₂₈) = 37 − 28 / (5 − 1) = 30
   12₃₆ = 19₃₆ − (10₃₆ − 1₃₆) / 5₃₆ = 45 − (36 − 1) / 5 = 38

(Yes, I know that unmarked numbers are in base 10, but it looks as if this question isn't solvable with a bit of lateral thinking.)

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  • $\begingroup$ I concur. It is not solvable in base-10. But since base is not specified... $\endgroup$
    – Overmind
    Commented Apr 12, 2017 at 11:47
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Are you sure, the numbers are 1,5,19,10? I guessing answer was (19/5)+10-1 = 12.8 (or) (5/19)+1*10 = 12.6 by following under conditions (But it's not equal to 12).

we can do like $(5+1+10) - \sqrt{(19+1)/5} = 16-4.36 = 11.64$ (not equal to 12).

I believe $10 + \sqrt{(19+1)/5} = 12$ (need to use Square root)

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