NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.
I'll get to that cryptic title in a moment (may be obvious to some)...
1. Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:
Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc... You may combine the digits to form numerals like 12 or 120.
Mr. Lomas: 5! - (2+2)! - 1 - 0 Perhaps the most elegant since it uses the individual digits in the given order.
Robot Guy: (21-2)*5+0
Nate (high schooler): 120-5^2 Oh, the simplicity of that one! Combining digits is not the first way I thought of...
Mine so far:
(a) 102 - (5+2) Pretty simple but I wasn't thinking much of combining digits until I saw Nate's
(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)
(c) (2^5)(2+1) - 0! (I posted this one already)
(d) 10^2 - 5 x (2 - 0!) (I knew there had to be a way using 100 - 5)
NOTE: JONATHAN POINTED OUT MY ERROR HERE. SEE COMMENTS.
(e) A new one: (2 + 2)! x (5-1) - 0! I felt I needed to atone for my error in (d)!
I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!
Keep these coming if you can find more. I'd like to see us get to 10 ways.
(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.
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7 comments:
(0+1)| (2x2)!-5! |
three of clubs--
I like that form but check the arithmetic. If (0+1) is multiplied by the absolute value then I obtain 96. I think all you need do is to take your expression in abs values and then subtract (0+1) yielding 96 - 1 = 95.
Regardless, you found a very interesting one.
Dave,
there's an extra zero in your example d.
Not very original, but with a twist:
[(2 - 0!)/.1]^2 - 5
Jonathan
jonathan--
thanks for pointing out my error!
Naturally your decimal approach makes me think of using repeating decimals:
2 / (.02 repeating) + 1 - 5 = 2 / (2/99) -4 = 99-4 = 95.
I owe that one to you Jonathan! I hope it makes up for my carelessness.
These kinds of number puzzles do take on a life of their own and there are so many popular variations of these kinds of problems. I really do think they improve student's number sense and review order of operations.
In actual practice, do you think most students would have calculator in hand while playing with this riddle? How many of your students do you think would attempt to do this with paper and pencil and some mental calculation like we do! I'm sure there are some...
I just added an extra one in the original post, labeled (e). Hopefully I've now atoned for "my sin"!! (wasn't My Sin once a perfume by some French company?).
Here it is again:
(5-1) (2+2)! - 0!
Is there any end to this? Actually, the issue of how many possible expressions one could make seems to be a formidable challenge, even if we place restrictions on the operations.
I do believe there are limits if you restrict the operations adequately.
With just + - * / ^, I don't think there are any others besides
120 - 5^2
(21+0-2)*5
5*(21-2+0)
(2-21) * (0-5)
etc.
102 - (5+2)
102 - 5 - 2
etc.
(50-2)*2-1
In terms of limits in general, with binary operations you always consume an input, so the search depth is limited by that. If you add in unary operations like sqrt or factorial, then you can deepen quite a lot more, though perhaps not usefully. Other unary operations like floor and ceiling only apply once non-redundantly, so those don't matter.
Other binary operations I'd try would include modulo, log with a specified base, nth-root... what else?
Grem
Grem--
Nice analysis! I knew one of my readers would be able to see a few levels into this. Along the way, your systematic approach produced a few other nice combinations. Thanks! Now how would you write a program in Python to analyze all the possibilities!
By the way, I'm sure I could have picked almost any other date and someone's age and we could have generated many solutions. But I just get the feeling there's something special about Martin Gardner's legacy to us...
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