Integers don't always do what you want them to do, unless you change the rules
The numerals on a digital clock are each made up of some combination of seven line segments, as shown above. The numeral 8, for instance, requires all seven segments. 1.
Look at the 10 digits displayed above and note that the digit 4 is composed of four segments. Which other digits accurately name the number of segments they contain? 2.
Instead of using seven individual segments, we can use the segment combinations shown at right to create all 10 digits. You can turn, flip, and overlap the combinations as needed. For example, one way to build the numeral 8 is to use shapes 2, 4, and 5. But you don't need all six patterns. Which can you discard and still be able to make all 10 digits? (You may use each segment combination only once in each digit.) 3.
When a digit of the clock changes, some segments turn on and some turn off. When 4 turns to 5, for instance, two segments turn on and one segment turns off, for a total of three changes. What is the largest number of segments that change during a transition from one digit to the next? What is the smallest number of segments that change? (Don't forgetin the inexorable passage of time, 9 changes to 0.) 4.
During the 10 transitions through all the digits, there are a total of 30 segment changes. Can you put the 10 digits in a different order so that the 10 transitions, including the change from the last digit to the first, require only 16 segment changes?
When One Plus One
Doesn't Equal Two
Numbers aren't particularly self-reflective. In English, there's only onethe number fourthat has the same number of letters as its name. But we can make number expressions that describe the number of letters they contain. TWO + FIVE, for instance, adds up to seven and has a total of seven letters. Can you complete the following self-counting expressions? Each question mark stands for a single digit (0 through 9), and digits cannot be repeated within an expression. 1.
TWO + FIVE = 7 2.
THREE + ? 3.
FIVE + SIX + ? 4.
NINE + TWO + ? 5.
SIX + SEVEN + ? + ? 6.
ZERO + ONE + TWO + ? + ? 7.
? + ? + ? + ? + ? + ? + ? + ? + ? 8.
What number in English, when written entirely with straight lines, represents the number of line segments it contains? For example, FIVE,
which does not describe itself, contains 10 line segments.
Sometimes the wrong method can turn out a correct answer. For example, suppose you are trying to simplify the fraction 16/64. If you have even a glancing acquaintance with arithmetic, you know that canceling out the two 6s is not the way to go about it. But the resulting fraction, 1/4, is indeed the right answer to the problem.
It's just a coincidence, of course. The technique won't usually work. However, the eight equations below do yield the correct simplified fraction if you cancel similar digits in the numerator and the denominator. Can you fill in the missing digits? Fractions in which all the digits are the same, such as 11/11 = 1/1, are not allowed. Solution
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