# Bogglers

By Scott Kim|Friday, September 01, 2000

Digital Dexterity
It's easy to make a circle by touching your left and right thumbs and left and right forefingers as shown here, but have you tried tying your fingers into knots? See if you can contort your digits to form the knots shown below. (Note: You need thin, flexible fingers to do figures 2 and 3; not everyone's fingers are long enough to reach.) 1.  { EASY } Joining thumbs to forefingers, can you make a figure eight as shown below? 2.  { MEDIUM } Using only your thumbs and forefingers, can you make an open figure eight, as shown below? 3.  { HARD } Using only your thumbs and forefingers, can you make a closed overhand knot, as shown at right? It's OK to distort the knot, as long as it stays connected in one continuous loop. Find two different solutions: one that joins thumbs to forefingers, and one that joins thumb to thumb and forefinger to forefinger. Scambled Geometry
{ MEDIUM } We've scrambled 12 words that you might remember from geometry class to form the quirky phrases below. Can you rearrange the letters of each phrase to make a familiar geometry term? Here are 12 hints to help you out (one hint per phrase, in no particular order): across the middle; it's all the same; around the edge; three-dimensional; divide and conquer; exaggerated curve; four faces; rickety rectangle; symmetrical triangle; the right stuff; the third side; triangle measurement. To get you started, the answer to the first phrase is parallelogram, and the hint is rickety rectangle.

 1. alarm galloper 2. barley hop 3. earth rodent 4. grey tin motor 5. hop dearly 6. nuclear dipper 7. prime tree 8. read time 9. relate quail 10. sees coils 11. to scribe 12. upset honey

Base Impulses

We might very well have our 10 fingers to thank for our use of the base 10 system. If we had 12 fingers, chances are we would count in base 12. In our base 12 world, digits 0 through 9 would stay the same, but 10 and 11 would be represented by two "new" digits, let's say T and E. And instead of the tens, hundreds, and thousands places, we would have the 12s, 144s (12 x 12), and 1,728s places (12 x 12 x 12). Note that from right to left, every successive place is 12 times greater than the previous place.

In general, the place values in any base n to the left of the decimal point are the ones place, the n's place, the n squared place, the n cubed place, and so on. The place values to the right of the decimal point are the 1/n place, the 1/n squared place, and so on. To compute the total value of a number in base n, we multiply each digit by its place value and add up all the products.

1.  { EASY } How would you write the number 1,967 in base 12?

2.  { EASY } In base 10, the fraction 1/9 is equal to the repeating decimal fraction .1111...  How would you write 1/9 as a fraction in base 12?

3.  { MEDIUM } Here's a weird one. In base -10 we count using the 10 digits 0 through 9, in which we have the ones place, the negative tens place, the positive hundreds place (-10 x -10), the negative thousands place (-10 x -10 x -10), and so on. How would you write 1,967 in base -10? There are actually two ways of writing every number in base -10. Can you find the other way of writing 1,967?

4.  { MEDIUM } How would you write 1,967 in base 1/10?

5.  { HARD } How would you write 1,967 in base 10 if, instead of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, we used the digits -5, -4, -3, -2, -1, 0, 1, 2, 3, and 4?

Solution

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