Birthday Cards

[Ruff_Hi]

There was a boy who was born in 1973. His mathematical uncle sends him a birthday card each year that contains a formula that uses the digits 1, 9, 7 and 3 (in order) and the mathematical symbols '+ - x / SQT ^ ! ( )'.

If both live until the boy turns 100, which ages would foil the uncle's birthday card plans?

[Ruff_Hi]

I'm actually working on the answers to this myself and I have solutions for 97 of the 100 ages. The ones 'missing' are: 92, 93 and 95. Extra gold star or <img src=/S/ribbon.gif border=0 alt=ribbon width=15 height=15> for the people who contribute those ages!

[HansV]

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>92: 1+97-3!
93: -1+97-3
95: 1+97-3</font color=yellow></span hi>

[Ruff_Hi]

Now why didn't I think of those. I was constucting answers that had at least 1 mathematical symbol between each number and so putting <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow> the 97 in </font color=yellow></span hi> like that didn't even occur to me - duh!

[Tricky]

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>I'm guessing that ! is a mathematical symbol? I see that it is in Timbo's list of useful items but I don't recall using this before for anything other than WOW!</font color=yellow></span hi> <img src=/S/confused.gif border=0 alt=confused width=15 height=20>

[HansV]

The ! stands for factorial. n! or n factorial is the number of ways you can arrange n distinct objects in a row. Say for example that you have three cards: a red one, a green one and a blue one. In how many ways can you arrange them in a row next to each other?

red - green - blue
red - blue - green
green - red - blue
green - blue - red
blue - red - green
blue - green - red

That makes 6. You can compute it as follows: work from left to right. For the first card, you have 3 options. After that, you have 2 options left for the second card. The third card is determined by the first two, so you have only one option. The number of combinations is 3 x 2 x 1 = 6.

1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120

etcetera. In general n! = n x (n-1)!
These numbers grow very fast.

[Tricky]

That must've been covered in class while I was snoozing in the corner. Thanks for the explanation. <img src=/S/cool.gif border=0 alt=cool width=15 height=15>

[Ruff_Hi]

Hans,

Actually, I don't think that the factorial (!) function does represent the number of ways of arranging items. The mathematical function 'factorial' is n x (n-1) x (n-2) x ... x 1. It just happens to give the 'number of ways of arranging' answer.

More generally, permutation represent the number of distinct ways of arranging n items from a collection of p. The answer is p! / (p-n)! So, if you want the number of permutations of 3 numbers from 1, 2, 3 and 4 then the answer is 4! / (4-3)! = 24 / 1 = 24. It is easier (sometimes) to think of this as 'I have 4 choices for the first number, 3 for the second and 2 for the third' so the answer is 4 x 3 x 2 = 24.

Now combinations represent the number of non-distinct ways of arranging n items from a collection of p. The answer is p! / (p-n)! n! So, if you want 3 combinations of numbers from 1, 2, 3 and 4 then the answer is 4! / (4-3)! 3! = 24 / 1 / 6 = 4. Using the simple example above, 'I have 4 choices for the first number, 3 for the second and 2 for the third but once I have my numbers, there are 3 x 2 x 1 ways of arranging them' so the answer is 4 x 3 x 2 / (3 x 2 x 1) = 4

The difference between a permutation and a combination is that 432 and 234 are different permutations but they are NOT different combinations.

This is a branch of probability mathematics and helps to answer questions like 'how many bridge hands can i draw?' A: 52! / 13! / 39! = 635,013,559,600 - a rather large number.

I have even come across a table of non integer factorials - say factorial of 0.5. Once you have that, you can also get the factorial of 1.5, 2.5, 3.5, etc using the function that Hans mentioned. The memory is going so I cannot now remember what you might use these non integer factorials for.

[HansV]

You're correct that n! is the number of permutations of n distinct items. My reply did not use (was not intended to use) exact mathematical terminology.

The factorial function is a special case of the gamma function; to go into it deeply is far beyond the scope of the Puzzles forum. If you're not afraid of formulas take a look at Factorial and Gamma Function on the MathWorld site.

[Ruff_Hi]

Ah - that's it - the gamma function - takes me all the way back to 2nd year uni where we studied that in one class and the prof sprung this function in the final test in another class. People that didn't take both classes were complaining that they were disadvantaged. The question - prove that gamm(n) = n! or something similar.

[AlanMiller]

I suspect it was to prove gamma(n) = (n-1)!
I also seem to recall similar headaches with the topic being "brushed over" but then examined.

Alan

[Ruff_Hi]

I've been informed that excel gives an incorrect answer for FACT(1.5). It treats it the same as FACT(1) which is incorrect. The issue for me is that 20 of my 100 ages are now 'wrong' - sort of back to square one.

Also, it was to prove that gamma(n) = (n-1)! given that the gamme formula is as shown below. Ok, it was a relatively new function (maybe completely new) but it was easy to do using the proof by induction method.

[AlanMiller]

Now there's a bit of nostalgia - mathematical induction. That's one I could never really get a grip on - not the method itself, but identifying whether a postulate was an appropriate candidate for the method. I guess it's relevant only for integer (ordinal) expressions, but beyond that ???? Maybe Hans has some insights?

Alan

[HansV]

The Excel FACTworksheet function is not intended to implement the Gamma function. The online help says

<hr>FACT(number)

Number is the nonnegative number you want the factorial of. If number is not an integer, it is truncated.<hr>

So the behavior you see is by design. The GAMMALN worksheet function returns the natural logarithm of the gamma function, e.g. EXP(GAMMALN(5)) is the same as FACT(4), and to compute FACT(1.5), you can use EXP(GAMMALN(2.5)). However, I don't think the maker of the puzzle intended to allow using the gamma function.

[Ruff_Hi]

Here are the 13 'ages' that I am missing - help please!

41, 42, 54, 55, 56, 58, 84, 85, 86, 87, 88, 97, 98