1+1=10
Despite what the title might lead you to believe, this article isn't a cover letter to the Trump people expressing my interest in joining their team as an economic or tax advisor. The title is sound math.
The savvy reader will recognize that "one plus one equals ten" is not how the title is read. Thanks to computers, most of us (probably those under forty, at least) would recognize the title as machine language, binary code that is read as "one plus one equals one zero." The number that looks like a ten is not a ten and is not called a ten. The simple math equation is a perfect example of what math is.
Math is a set of rules that allows us to count and measure things using symbols. The basic symbols we use are numbers, each number representing a quantity that we can easily count on our hands. One finger became the symbol, 1. It's not a coincidence the symbol to represent one finger looks like a finger. Two should look like the peace sign or two fingers, but somehow ended up looking like 2 instead. Three should have looked like the a-ok sign, but ended up looking like 3 instead. The long and short of it is we created a symbol to represent each of our fingers so we could communicate with others how many of something there was.
Eventually, we took the simple symbols we used for counting, invented a new one to represent nothing, and our number system in Base 10 (Base 10 because we have ten fingers) was born. Some really smart people assigned rules to how we use the symbols because, while we all could count to ten using our fingers, we needed an easy way to count past ten. Rules of addition and subtraction were developed and later, multiplication and division - which are faster ways to add and subtract. Long story short, math (in Base 10) saves me the embarrassment of taking my shoes off in the liquor store to count twenty cents change to the cashier...twenty-one if I get naked. If I had to count past twenty-one, I would have to take the beer and run. Then I would be faced with the impossible task of counting 3,650 days until the end of my ten-year sentence.
Did you catch how I manipulated the explanation to introduce the concept of more complex symbols to represent batches of number symbols and used rules to make counting the symbols easier?
Easy, right? But there are a lot of things about math we take for granted as the perception puzzle disguised as a math problem on the right proves. Can you solve it by applying rules to the symbols? Feel free to click on the picture to view a larger version. I don't want someone coming back with the wrong answer and complaining that if I had posted a larger picture, they'd have been able to solve it.
Did you solve it? I'll give you a hint. The answer is not greater than 14 nor is the answer less than 14. Yes, I cleverly introduced new rules to manipulate the symbols, greater and less than. The answer to the puzzle is 14.
You can click on the link under the picture to read their explanation, but it might leave you more confused. At least on the forum where I ran across this puzzle, a couple of people refused to accept 14 as the answer, probably because that wasn't the answer they guessed, and proceeded to "prove" the puzzle was poorly designed and misleading.
Since symbols in math have to remain consistent, an apple wherever it appears in the puzzle must equal one number. By the rule of division, the top line dictates that each apple must equal 10. The second line is asking what number added to itself equals 8, since we know the apple is 10. The grapes must equal 4. The third line is asking what number subtracted from 4 (the value of the grapes) equals 2. The two half coconuts, therefore must equal two.
The last line is the perception trick. From the line above, we know two half coconuts equal two so it follows one half coconut must equal 1. 1 (half coconut) + 10 (apple) + 4 (grapes) = 15.
"Hold on a minute!" you might be thinking. "I thought you said the answer was 14."
Look at the grapes again. One grape is missing from the bunch. That means the bunch of grapes on the last line can't equal 4. Just as we looked at two half coconuts as equaling two, we can look at the bunch of grapes and deduce that the number of grapes in the bunch determines the numeric value of the bunch. The bunch of grapes on the last line is missing a grape. It, therefore, must equal 3. The equation can now be more accurately written as 1 (half coconut) + 10 (apple) + 3 (grapes) = 14.
The biggest argument on the forum was that the puzzle was poorly designed. Since an apple equals 10, why couldn't the other produce equal something arbitrary, too. "It makes no sense!" the objections resounded.
No, it makes sense if we take nothing for granted. Math is a universal language, but you have to know the base you're using the language in. We operate in Base 10 probably because we have ten fingers. An eight-tentacled alien with ten suction cups on each tentacle might be operating in Base 80. But if we know what base they are working in, we can communicate with them because the math rules apply the same in Base 80 as they do in Base 10.
"Huh?" you ask?
Look at the title of the article, again. 1+1=10 makes no sense in Base 10, but is mathematically correct in Base 2. Computers only understand two states: on or off. When we communicate with machines, we have to communicate in Base 2. We take our decimal code and convert it to binary code so the machine understands us. The machine does its calculations in binary code, then converts it to decimal code so we understand the machine.
When the symbols we use are Arabic numbers, we can guess what base we're working in. If there are only 0's and 1's, we're in Base 2. 0's through 9's, Base 10. 0's through F, Base 16. I have no clue what Base 80 would look like. I reckon I would come across as a jibbering idiot to the eight-tentacled alien.
Our math problem, however, doesn't use Arabic numbers. It uses produce. I hate to say fruit because someone would be quick to point out coconuts and grapes aren't fruits. Hopefully produce is safe to say since they all can be found in the produce section of your local grocery store.
That said, without the numeric cues, how would we know what mathematical base this problem is working in? Apples are fruit. (They really are, botanically speaking.) Fruit tell us nothing, mathematically. And that's where our assumptions we take for granted gets us in trouble solving this problem.
The designers of this perception problem had to tell you what base the problem is to be solved in. Since the designers used produce to represent numbers, our first error in solving the problem would be to assume it's to be solved in Base 10.
If the three apples equaled 11, we know the rest of the problem is solved in Base 2. If the three apples equaled 10, we know the rest of the problem is solved in Base 3. Since the three apples equal 30, we know the problem is solved in Base 10, the system we're most comfortable with. If the three apples, however, equaled x, we might be looking at an algebra problem or a Base 23 math problem. I won't complicate the explanation any further than I have, but if you want to know why three apples equaling x could be an algebra problem or a Base 23 problem, suffice it to say it's how mathematicians defined conventions...not rules, but conventions.
For those who answered the problem incorrectly, but always have to be right, no, it doesn't matter two halves of a coconut equals two and one half of a coconut equals one. Remember, math is a set of rules telling us how to count. The first line defines the base we are to solve the problem in. The next two lines define what we are counting. The two halves of the coconuts tells us we're counting pieces of produce as one. We aren't worried about fractions. So the grape bunch with only 3 grapes isn't three-quarters. It's 3.
And for those who insist they aren't grapes; they're prunes, ok. Either I need a botany class or the designers of the problem need to hire better artists.
TL;DR Folks:
This article solves a basic math problem. Until you get to calculus and combinatorics, there are no shortcuts. You'll have to solve the problem the good old fashioned way, complete with the long explanation when you want to know why you got the wrong answer.
Posted by A Drunk Redneck
The savvy reader will recognize that "one plus one equals ten" is not how the title is read. Thanks to computers, most of us (probably those under forty, at least) would recognize the title as machine language, binary code that is read as "one plus one equals one zero." The number that looks like a ten is not a ten and is not called a ten. The simple math equation is a perfect example of what math is.
Math is a set of rules that allows us to count and measure things using symbols. The basic symbols we use are numbers, each number representing a quantity that we can easily count on our hands. One finger became the symbol, 1. It's not a coincidence the symbol to represent one finger looks like a finger. Two should look like the peace sign or two fingers, but somehow ended up looking like 2 instead. Three should have looked like the a-ok sign, but ended up looking like 3 instead. The long and short of it is we created a symbol to represent each of our fingers so we could communicate with others how many of something there was.
Eventually, we took the simple symbols we used for counting, invented a new one to represent nothing, and our number system in Base 10 (Base 10 because we have ten fingers) was born. Some really smart people assigned rules to how we use the symbols because, while we all could count to ten using our fingers, we needed an easy way to count past ten. Rules of addition and subtraction were developed and later, multiplication and division - which are faster ways to add and subtract. Long story short, math (in Base 10) saves me the embarrassment of taking my shoes off in the liquor store to count twenty cents change to the cashier...twenty-one if I get naked. If I had to count past twenty-one, I would have to take the beer and run. Then I would be faced with the impossible task of counting 3,650 days until the end of my ten-year sentence.
Did you catch how I manipulated the explanation to introduce the concept of more complex symbols to represent batches of number symbols and used rules to make counting the symbols easier?
 |
| This quiz and others can be found at Brain Candy |
Did you solve it? I'll give you a hint. The answer is not greater than 14 nor is the answer less than 14. Yes, I cleverly introduced new rules to manipulate the symbols, greater and less than. The answer to the puzzle is 14.
You can click on the link under the picture to read their explanation, but it might leave you more confused. At least on the forum where I ran across this puzzle, a couple of people refused to accept 14 as the answer, probably because that wasn't the answer they guessed, and proceeded to "prove" the puzzle was poorly designed and misleading.
Since symbols in math have to remain consistent, an apple wherever it appears in the puzzle must equal one number. By the rule of division, the top line dictates that each apple must equal 10. The second line is asking what number added to itself equals 8, since we know the apple is 10. The grapes must equal 4. The third line is asking what number subtracted from 4 (the value of the grapes) equals 2. The two half coconuts, therefore must equal two.
The last line is the perception trick. From the line above, we know two half coconuts equal two so it follows one half coconut must equal 1. 1 (half coconut) + 10 (apple) + 4 (grapes) = 15.
"Hold on a minute!" you might be thinking. "I thought you said the answer was 14."
Look at the grapes again. One grape is missing from the bunch. That means the bunch of grapes on the last line can't equal 4. Just as we looked at two half coconuts as equaling two, we can look at the bunch of grapes and deduce that the number of grapes in the bunch determines the numeric value of the bunch. The bunch of grapes on the last line is missing a grape. It, therefore, must equal 3. The equation can now be more accurately written as 1 (half coconut) + 10 (apple) + 3 (grapes) = 14.
The biggest argument on the forum was that the puzzle was poorly designed. Since an apple equals 10, why couldn't the other produce equal something arbitrary, too. "It makes no sense!" the objections resounded.
No, it makes sense if we take nothing for granted. Math is a universal language, but you have to know the base you're using the language in. We operate in Base 10 probably because we have ten fingers. An eight-tentacled alien with ten suction cups on each tentacle might be operating in Base 80. But if we know what base they are working in, we can communicate with them because the math rules apply the same in Base 80 as they do in Base 10.
"Huh?" you ask?
Look at the title of the article, again. 1+1=10 makes no sense in Base 10, but is mathematically correct in Base 2. Computers only understand two states: on or off. When we communicate with machines, we have to communicate in Base 2. We take our decimal code and convert it to binary code so the machine understands us. The machine does its calculations in binary code, then converts it to decimal code so we understand the machine.
When the symbols we use are Arabic numbers, we can guess what base we're working in. If there are only 0's and 1's, we're in Base 2. 0's through 9's, Base 10. 0's through F, Base 16. I have no clue what Base 80 would look like. I reckon I would come across as a jibbering idiot to the eight-tentacled alien.
Our math problem, however, doesn't use Arabic numbers. It uses produce. I hate to say fruit because someone would be quick to point out coconuts and grapes aren't fruits. Hopefully produce is safe to say since they all can be found in the produce section of your local grocery store.
That said, without the numeric cues, how would we know what mathematical base this problem is working in? Apples are fruit. (They really are, botanically speaking.) Fruit tell us nothing, mathematically. And that's where our assumptions we take for granted gets us in trouble solving this problem.
The designers of this perception problem had to tell you what base the problem is to be solved in. Since the designers used produce to represent numbers, our first error in solving the problem would be to assume it's to be solved in Base 10.
If the three apples equaled 11, we know the rest of the problem is solved in Base 2. If the three apples equaled 10, we know the rest of the problem is solved in Base 3. Since the three apples equal 30, we know the problem is solved in Base 10, the system we're most comfortable with. If the three apples, however, equaled x, we might be looking at an algebra problem or a Base 23 math problem. I won't complicate the explanation any further than I have, but if you want to know why three apples equaling x could be an algebra problem or a Base 23 problem, suffice it to say it's how mathematicians defined conventions...not rules, but conventions.
For those who answered the problem incorrectly, but always have to be right, no, it doesn't matter two halves of a coconut equals two and one half of a coconut equals one. Remember, math is a set of rules telling us how to count. The first line defines the base we are to solve the problem in. The next two lines define what we are counting. The two halves of the coconuts tells us we're counting pieces of produce as one. We aren't worried about fractions. So the grape bunch with only 3 grapes isn't three-quarters. It's 3.
And for those who insist they aren't grapes; they're prunes, ok. Either I need a botany class or the designers of the problem need to hire better artists.
TL;DR Folks:
This article solves a basic math problem. Until you get to calculus and combinatorics, there are no shortcuts. You'll have to solve the problem the good old fashioned way, complete with the long explanation when you want to know why you got the wrong answer.
For your listening pleasure - Letting little ducks represent numbers:
Posted by A Drunk Redneck
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