Bogglers Solutions

By Scott Kim|Tuesday, May 01, 2001




The Toothpick Tally








Don't Count That Dial!

1) 144 area codes are possible.

2) 630 area codes are possible.

3) Answers: 28, 28, and 1,716. Shown below is a spreadsheet of the number of phone numbers of a particular length that add up to a particular value. To compute the value in any particular cell, add the values in the 10 cells starting from the cell directly above and moving left. For instance, the value in the green cell below is the sum of the values in the orange cells. (There are fewer than 10 orange cells because the eighth, ninth, and tenth cells fall off the left edge of the table.) This formula works because to make a three-digit number whose digits add up to 8, you start with a two-digit number whose digits add up to 2, 3, 4, 5, 6, 7, or 8 and append the appropriate digit. For instance, 41 is one of the four different two-digit numbers whose digits add up to 5. Append the digit 3 and you get 413, which is a three-digit number whose digits add up to 8. Mathematically inclined readers will recognize this table as a portion of Pascal's triangle; the values diverge from Pascal's triangle once the sum exceeds 11.

# of
digits
Sum of Digits
2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
1
1
1
1
1
1
1
1
2
3
4
5
6
7
1
3
6
10
15
21
28
1
4
10
20
35
56
84
1
5
15
35
70
126
210
1
6
21
56
126
252
462
1
7
28
84
210
462
924
1
8
36
120
330
792
1,716





Counting on Your Fingers

1) 11: the numbers 0 through 10.

2) 100. Each hand can count 10 different numbers.

3) 1,024. This is counting in base 2. You may enjoy counting from 0 to 31 in base 2 on one hand— it's quite a finger twister.

4) 1,546. There is one way for no fingers to touch, and 5 x 5 = 25 ways for one finger of one hand to touch one finger of the other hand. There are 10 ways to pick two fingers of one hand and therefore 10 x 10 x 2 = 200 ways to touch two fingers of one hand to two fingers of the other hand. The extra "x 2" accounts for the possibility that the order of fingers on one hand is reversed. Continuing to count the number of ways to touch three, four, and five pairs of fingers together, we get 1 + 25 + 200 + 600 + 600 + 120 = 1,546.






Want to go back to the puzzle?

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.


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