GC156TK One Million Factorial (Unknown Cache) in Washington, United States created by Camp Latona

GC156TK ▼

A cache by Camp Latona Message this owner

Hidden : 8/16/2007

Difficulty:

Terrain:

Size: Size: micro (micro)

Join now to view geocache location details. It's free!

Watch

How Geocaching Works

Please note Use of geocaching.com services is subject to the terms and conditions in our disclaimer.

Geocache Description:

Here's a puzzle for mathematically minded geocachers.

1000000! = 1 * 2 * 3 * 4 ... * 999998 * 999999 * 1000000

The equation above results in a gigantic number with over 5.6 million digits. Any of the multipliers that include factors of five add zeros to the end of result. Overall, almost a quarter of a million trailing zeros are generated.

To solve the puzzle, find the six rightmost non-zero digits of 1000000!. These are the ones that come just before the trailing zeros begin. Call these digits abcdef. Adding the digits and re-adding until a single digit remains, gives a digital root of 2.

Calculate the final coordinates as:
N47 40.[e - c] [b + c] [f - b], W122 18.[e - b] [a + b] [c - b]

When calculating the coordinates:
If a result exceeds 9, use the rightmost digit only.
If a result is negative, flip the sign to positive.

This puzzle is obviously not for the uninitiated. The difficulty is not just that multiplying integers one by one (1*2=2, 2*3=6, 6*4=24, and so on) quickly exceeds your willingness to type. Using the factorial function (!) on a 10-digit scientific calculator can only display results exactly up to 13!. It can display approximate values for larger factorials by using scientific notation. After 69! that even that fails, as the exponent exceeds the allowed 2 digits. Computer calculators offer a little more help because larger word sizes allow more digits to be displayed. But they also fail surprisingly far short of the goal.

On-line java applets, like BigNumCalc are a big step forward. This one can display 1000! exactly, at least to the point where the trailing zeros begin. Somewhere, as these larger and larger factorials are calculated by BigNumCalc, digits are truncated from the end of the result until all of the trailing zeros disappear and finally significant digits are lost. After about 10000! it no longer is even willing to try, displaying “Factorial out of range.” So, trying to compute all 5.6 million digits is a tough way to approach this puzzle.

Sites like mathworld offer some math background and this one offers advice to a math student about how difficult it is even to compute an approximate value, but then proceeds to do so.

Luckily, to find the six digits you don't need to find all 5.6 million. Surprisingly, it’s not as hard as it may seem at first. All that is needed is a simple algorithm and a tiny computer program to execute it. The need for programming may exclude some of you, but perhaps those that don't program can phone a friend who does. Or, if your math skills are up to it, you might be able to find another way. Let's see who among you can crack it.

Tip: To find 6 digits correctly my program had to compute 7 dights.

Hide: Use hint below to overcome poor GPS reception.

Additional Hints (Decrypt)

gjb-guveqf gur jnl qbja uvyy orgjrra gjragl bu-gjb naq gjragl bu-sbhe.

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)

Watch

Geocache Description:

Attributes

Inventory

Bookmark Lists

One Million Factorial Mystery Cache

Watch

Geocache Description:

Attributes

Inventory

Bookmark Lists

Treasures