A minor diversion
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- Dalva Golden White Colheita 1952
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A minor diversion
Using only the numbers 1 to 4, with each being used as a single digit once in each case, can the numbers 1 through 40 be created using only these numbers plus mathematical symbols? e.g. 21 = (1+2).(3+4)
Note: (1) All numbers 1-4 must be used in each case, and each must be distinct (using them as numbers, not digits, so e.g. 34-21 is not allowed), and (2) Warning - the answer may be no! (I don't know if all are possible myself, yet, though I know most are).
Note: (1) All numbers 1-4 must be used in each case, and each must be distinct (using them as numbers, not digits, so e.g. 34-21 is not allowed), and (2) Warning - the answer may be no! (I don't know if all are possible myself, yet, though I know most are).
Re: A minor diversion
Can 4 be used as a 2 simply by including (√4) in an equation?
or is it only +, -, x, /, ^ that can be used?
or is it only +, -, x, /, ^ that can be used?
Rob C.
Re: A minor diversion
1 = (1+4)/(2+3)
2 = 1+2+3-4
3 = (3+4-1)/2
4 = ((1+3)x2)-4
5 = ((1+2)x3)-4
6 = ((3x4)/2)x1
7 = ((4+1)x2)-3
8 = 2-1+3+4
9 = (3x4)-(1+2)
10 = ((3x4)-2)x1
11!back later
2 = 1+2+3-4
3 = (3+4-1)/2
4 = ((1+3)x2)-4
5 = ((1+2)x3)-4
6 = ((3x4)/2)x1
7 = ((4+1)x2)-3
8 = 2-1+3+4
9 = (3x4)-(1+2)
10 = ((3x4)-2)x1
11!back later
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Ernest H. Cockburn
Ernest H. Cockburn
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Re: A minor diversion
11 = 3x4-2+1DRT wrote:1 = (1+4)/(2+3)
2 = 1+2+3-4
3 = (3+4-1)/2
4 = ((1+3)x2)-4
5 = ((1+2)x3)-4
6 = ((3x4)/2)x1
7 = ((4+1)x2)-3
8 = 2-1+3+4
9 = (3x4)-(1+2)
10 = ((3x4)-2)x1
11!back later
12...
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- Dalva Golden White Colheita 1952
- Posts: 3522
- Joined: 14:22 Wed 15 Dec 2010
- Location: Near Cambridge, UK
Re: A minor diversion
The √ symbol is acceptable (though possibly not much help).RAYC wrote:Can 4 be used as a 2 simply by including (√4) in an equation?
or is it only +, -, x, /, ^ that can be used?
Re: A minor diversion
Depressingly this "minor" diversion seems more exciting than knuckling down to work, and has become a major diversion!
12.(4x3)/(2-1)
13. 4x3+2-1
14. 4x3x1+2
15. 4x3+2+1
16. 4x(3+(2-1))
17. (4+1)x3+2
18. 4x(3+1)+2
19. 4x(3+2)-1
20. (2^4)+3+1
21. (4+2+1)x4
22. ((4+1)^2)-3
23. ((2+1)^3)-4
24. √((3+2)^4)-1
25. √((3+2)^4)x1
26. √((3+2)^4)+1
27. ((2+1)^4)/3
28. ((4+1)^2)+3
29. (2^(4+1))-3
30. 2x3x(4+1)
31. ((2+1)^4)+4
32. (2^4)x(3-1)
33. ((4^3)/2)+1
12.(4x3)/(2-1)
13. 4x3+2-1
14. 4x3x1+2
15. 4x3+2+1
16. 4x(3+(2-1))
17. (4+1)x3+2
18. 4x(3+1)+2
19. 4x(3+2)-1
20. (2^4)+3+1
21. (4+2+1)x4
22. ((4+1)^2)-3
23. ((2+1)^3)-4
24. √((3+2)^4)-1
25. √((3+2)^4)x1
26. √((3+2)^4)+1
27. ((2+1)^4)/3
28. ((4+1)^2)+3
29. (2^(4+1))-3
30. 2x3x(4+1)
31. ((2+1)^4)+4
32. (2^4)x(3-1)
33. ((4^3)/2)+1
Rob C.
Re: A minor diversion
DRT wrote:1 = (1+4)/(2+3)
2 = 1+2+3-4
3 = (3+4-1)/2
4 = ((1+3)x2)-4
5 = ((1+2)x3)-4
6 = ((3x4)/2)x1
7 = ((4+1)x2)-3
8 = 2-1+3+4
9 = (3x4)-(1+2)
10 = ((3x4)-2)x1
12 = (2x4)+1+3AHB wrote:11 = 3x4-2+1
13 = (3x4)+2-1
14 = ((3x4)+2)x1
15 = (3x4)+2+1
16 = (2+3-1)x4
17 = (4²+3-2)x1
Pause: is the use of the mathematical symbol ² permitted?
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn
Re: A minor diversion
We need to find a way to avoid two or more people wasting time simultaneously
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn
Re: A minor diversion
It feels wrong to me, but if you are allowed to use √ then it is hard to argue against it i suppose!!DRT wrote:Pause: is the use of the mathematical symbol ² permitted?
39 then becomes very easy indeed... (3²)x4+2+1
Phil to adjudicate - i am happy to remove the use of √
Rob C.
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- Dalva Golden White Colheita 1952
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- Joined: 14:22 Wed 15 Dec 2010
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Re: A minor diversion
That would count as use of the digit 2. In the same manner, 2√ or 3√ could be used for square root or cube root, thereby using the digit 2 or 3 as well as a 'symbol'.DRT wrote:DRT wrote:1 = (1+4)/(2+3)
2 = 1+2+3-4
3 = (3+4-1)/2
4 = ((1+3)x2)-4
5 = ((1+2)x3)-4
6 = ((3x4)/2)x1
7 = ((4+1)x2)-3
8 = 2-1+3+4
9 = (3x4)-(1+2)
10 = ((3x4)-2)x112 = (2x4)+1+3AHB wrote:11 = 3x4-2+1
13 = (3x4)+2-1
14 = ((3x4)+2)x1
15 = (3x4)+2+1
16 = (2+3-1)x4
17 = (4²+3-2)x1
Pause: is the use of the mathematical symbol ² permitted?
Re: A minor diversion
√ by itself is the symbol for square root, no?
No need to put a 2 in front...
But happy to remove use.
No need to put a 2 in front...
But happy to remove use.
Rob C.
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- Dalva Golden White Colheita 1952
- Posts: 3522
- Joined: 14:22 Wed 15 Dec 2010
- Location: Near Cambridge, UK
Re: A minor diversion
Indeed, though optionally you can put any digit in front to represent the n-th root - which in this case counts as using the digit, just as using the digit within a power would whether you use the form 3² or 3^2. Each digit must be used individually (i.e. not compounds such as 13) and if you can see the digit in the written expression, then it's been used.RAYC wrote:√ by itself is the symbol for square root, no?
No need to put a 2 in front...
Re: A minor diversion
OK. So my solution to 17 is void, but replaced by Rob's.
Rob's solution to 31 uses two 4s and no 3.
Rob's solution to 31 uses two 4s and no 3.
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn
Re: A minor diversion
It also makes 85, not 31.DRT wrote:Rob's solution to 31 uses two 4s and no 3.
It should be ((2+1)^3)+4.
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn
Re: A minor diversion
yes - mis-typed.DRT wrote:It also makes 85, not 31.DRT wrote:Rob's solution to 31 uses two 4s and no 3.
It should be ((2+1)^3)+4.
24-26 also need to be re-arranged in light of the conversation above re: use of square root.
I have 31,32,35,36,37 and 40 done...but 34 and 39 are troubling me!
Rob C.
Re: A minor diversion
Y'all are making some of these way more complicated than necessary...
10: 1+2+3+4
23: 2*3*4-1
24: 1*2*3*4
25: 2*3*4+1
10: 1+2+3+4
23: 2*3*4-1
24: 1*2*3*4
25: 2*3*4+1
Glenn Elliott
Re: A minor diversion
I can get 39, but I can't express it with a keyboard.
Use sigma to express:
Sum(1..(4*2))+3
And following from that, 34 becomes:
Sum(3..(4*2))+1
Use sigma to express:
Sum(1..(4*2))+3
And following from that, 34 becomes:
Sum(3..(4*2))+1
Glenn Elliott
Re: A minor diversion
Here we go. Still not entirely sure what the "proper" notation is with a keyboard, but at least I can express sigma now.
39: 1∑(4*2)+3
34: 3∑(4*2)+1
Or perhaps...
39: ∑(1..4*2)+3
34: ∑(3..4*2)+1
39: 1∑(4*2)+3
34: 3∑(4*2)+1
Or perhaps...
39: ∑(1..4*2)+3
34: ∑(3..4*2)+1
Glenn Elliott
Re: A minor diversion
Yes...can't say i would have got there myself!Glenn E. wrote:Here we go. Still not entirely sure what the "proper" notation is with a keyboard, but at least I can express sigma now.
39: 1∑(4*2)+3
34: 3∑(4*2)+1
Or perhaps...
39: ∑(1..4*2)+3
34: ∑(3..4*2)+1
Rob C.
Re: A minor diversion
Like the best swiss watches...!Glenn E. wrote:Y'all are making some of these way more complicated than necessary...
Rob C.
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- Dalva Golden White Colheita 1952
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- Joined: 14:22 Wed 15 Dec 2010
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Re: A minor diversion
A good show! Am slightly unsure on the sigma notation validity?
I would normally have expected limits above and below, such as
4*2
∑(n) +3
n=1
although this needs additional algebra letters, which is perhaps borderline on being allowed...
If ∑(1..4*2) is acceptable to mean the sum of the list of numbers represented then I would have to allow it.
I'll share my alternatives:
34 = 4! +3^2 +1
39 = (3!)^2 +4 -1
Use of factorial makes at least 1-52 possible, not sure how much higher.
I would normally have expected limits above and below, such as
4*2
∑(n) +3
n=1
although this needs additional algebra letters, which is perhaps borderline on being allowed...
If ∑(1..4*2) is acceptable to mean the sum of the list of numbers represented then I would have to allow it.
I'll share my alternatives:
34 = 4! +3^2 +1
39 = (3!)^2 +4 -1
Use of factorial makes at least 1-52 possible, not sure how much higher.
Re: A minor diversion
Not to say that these aren't impressive, but summation and factorial operators are short-form definitions for much longer calculations (at least according to my very shaky understanding). So if square and square roots symbols can't be used without counting those as use of the number 2, i'm not convinced that these formulas can be used without counting all the numbers they involve when set out in long form!PhilW wrote:A good show! Am slightly unsure on the sigma notation validity?
I would normally have expected limits above and below, such as
4*2
∑(n) +3
n=1
although this needs additional algebra letters, which is perhaps borderline on being allowed...
If ∑(1..4*2) is acceptable to mean the sum of the list of numbers represented then I would have to allow it.
I'll share my alternatives:
34 = 4! +3^2 +1
39 = (3!)^2 +4 -1
Use of factorial makes at least 1-52 possible, not sure how much higher.
Rob C.
Re: A minor diversion
Perhaps this discussion has reached a stage where is should be posted here?
If anyone feels brave enough to step into that particular world of geekism
If anyone feels brave enough to step into that particular world of geekism
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn
Re: A minor diversion
From that place, a fabulous example of an entire discussion which neatly demonstrates the tendency of mathematicians to be brief and to the point!
"The first duty of Port is to be red"
Ernest H. Cockburn
Ernest H. Cockburn