Checkerboard Puzzle, Moore's Law, and Growth Prospects

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My father the mathematician first posed the checkerboard puzzle to me back in grade-school, perhaps on some rainy Saturday. His version of the story went something like this:

The jester performs a great deed, and the king asks him how he would like to be rewarded. The jester is aware that the king is a highly volatile individual, and if the jester asks for too much, the king might just kill him then and there. The jester also knows that the king views his promise as sacred, so if the king says "yes" to the jester's proposal, then the king will honor that promise. So in a way, the jester's problem is how to ask for a lot, but have the king at least initially think it's not very much, so that the king will give his consent.

So the jester clowns around a bit and then says: "Here's all I want. Take this checkerboard. On the first square, but one piece of gold. On the second square, two pieces. On the third square, four pieces, and on the fourth square, 8 pieces. Double the amount on each square until you reach the end of the checkerboard."

In the story, the king laughs at this comic proposal and says,  "Your great deed was so wonderful, I would have happily done much more than this! I grant your request!"

But of course, when the king starts hauling up gold pieces from the treasury, he will discover that 2 raised to the 63rd power, the final spot on the checkerboard requires about 9 quintillion gold pieces (that is, 9 followed by 18 zeros). 

I've had some sense of the power of exponential growth ever since.  But what I hadn't thought about is the interaction of Moore's Law and economic growth. Moore's Law is of course named for Gordon Moore, one of the founders of Intel, who noticed this pattern back in 1965. Back in the 1970s, he wrote a paper that contained the following graph showing how much it cost to produce a computer chip with a certain number of components. Here's his figure. Notice that the numbers of component on the horizontal axis and the cost figures on the vertical axis are both graphed as logarithm (specifically, each step up the axis is a change by a factor of 10). The key takeaway was that the number of transistors ("components") on an integrated circuit was doubling about every two years, making computing power much cheaper and faster.


Ever since I started reading up on Moore's law in the early 1980s, there have been predictions in the trade press that it will soon reach technological limits and come to and end. But Moore's law marches on: indeed, the research and innovation targets at Intel and other chip-makers are defined in terms of making sure that Moore's law continues to hold for at least awhile longer. Stephen Shankland offers a nice accessible overview of the current situation in an October 15, 2012, essay on CNET: ""Moore's Law: The rule that really matters in tech"  (The Gordon Moore graph above is copied from Shankland's essay.)

As Shankland writes: "To keep up with Moore's Law, engineers must keep shrinking the size of transistors. Intel, the leader in the race, currently uses a manufacturing process with 22-nanometer features. That's 22 billionths of a meter, or roughly a 4,000th the width of a human hair." He cites a variety of industry and research experts to the effect that Moore's law has at least another decade to run--and remember, a decade of doubling every two years means five more doublings!

It's hard to wrap one's mind around what it means to say that the power of microchipo technology will increase by a factor of 32 (doubling five times) in the next 10 years. A characteristically intriguing survey essay  from the January 10 issue of the Economist on the future of innovation uses the checkerboard analogy to think about the potential effects of Moore's law. Here's a comment from the Economist essay:

Ray Kurzweil, a pioneer of computer science and a devotee of exponential technological extrapolation, likes to talk of “the second half of the chess board”. There is an old fable in which a gullible king is tricked into paying an obligation in grains of rice, one on the first square of a chessboard, two on the second, four on the third, the payment doubling with every square. Along the first row, the obligation is minuscule. With half the chessboard covered, the king is out only about 100 tonnes of rice. But a square before reaching the end of the seventh row he has laid out 500m tonnes in total—the whole world’s annual rice production. He will have to put more or less the same amount again on the next square. And there will still be a row to go.

Erik Brynjolfsson and Andrew McAfee of MIT make use of this image in their e-book “Race Against the Machine”. By the measure known as Moore’s law, the ability to get calculations out of a piece of silicon doubles every 18 months. That growth rate will not last for ever; but other aspects of computation, such as the capacity of algorithms to handle data, are also growing exponentially. When such a capacity is low, that doubling does not matter. As soon as it matters at all, though, it can quickly start to matter a lot. On the second half of the chessboard not only has the cumulative effect of innovations become large, but each new iteration of innovation delivers a technological jolt as powerful as all previous rounds combined."

Now, it's of course true that doubling the capacity of computer chips doesn't translate in a direct way into a higher standard of living: there are many steps from one to the other. But my point here is to note that many of us (myself included) have been thinking about the changes in electronics technology a little too much like the king in the checkerboard story: that is, we think of something doubling a few times, even 10 or 20 times, and we know it's a big change, but it somehow seems within our range of comprehension.

But when something has already been doubling every 18 months or two years for a half-century--it is continuing to double!--the absolute size of each additional doubling is starting to get very large. I lack the imagination to conceive of what will be done with all this cheap computing power in terms of health care, education, industrial process, communication, transportation, entertainment, food, travel, design, and more. But I suspect that these enormous repeated doublings, as Moore's law marches forward in the next decade and drives computing speeds up and prices down, will transform lives and industries in ways that we are only just starting to imagine.

Taking Apprenticeships Seriously

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The United States puts a heavy emphasis on a college degree as the path to economic and social success, and thus it's a familiar pledge of politicians that a higher share of the population will attend college. For example, in a speech to Congress on February 24, 2009, President Obama 
set a goal that "by 2020, America will once again have the highest proportion of college graduates in the world."

But this emphasis on college has two difficulties: 1) as a society, we don't actually mean it; and 2) it probably isn't an appropriate goal, anyway. After all, if we really supported a widespread expansion of college education, we would do considerably more than pump up the loans available to students. Instead, we would be figuring out how current colleges can expand their enrollments, and starting a new wave of colleges and universities--and figuring out how to keep these options affordable to students. Instead, the U.S. has lost its lead as the country in the world with the highest proportion of college graduates.

Moreover, a four-year college degree just isn't going to be right for everyone. Think about those students who managed to finish a high school degree, but were in the bottom third or bottom quarter of the class. For many of these students, their interactions with the educational system have not been happy ones, and the notion that their life plan should start off with yet another four years of education is likely to be met with hard-earned dislike and disbelief.

So what's the alternative for these students, in a U.S. economy that places considerable value on skilled labor. Betty Joyce Nash offers one angle on these issues in in "Journey to Work: European Model Combines Education with Vocation  in the Fourth Quarter issue of Region Focus, which is published by the Federal Reserve Bank of Richmond.  She writes:

"In the United States, vocational education has been disparaged by some as a place for students perceived as unwilling or unable. The United States still largely champions college as the route to higher lifetime wages and the flexibility to retool skills in times of economic change. Yet just 58 percent of the 53 percent of college-goers in 2004 who started at four-year institutions finished within six years. Moreover, 25 percent of those who enter two-year community colleges don’t finish. Only about 28 percent of U.S. adults over age 25 actually have a bachelor’s. What about the rest? What’s their path to the workplace? It may be unrealistic to expect everyone to finish college, but most students will need more than a high school education as jobs become more complex."

Nash focuses her discussion on apprenticeships and vocational education, and as is common in these kinds of arguments, she focuses some attention on practices in Germany and Switzerland. Thus:

"Germany and Switzerland educate roughly 53 percent and 66 percent of students, respectively, in a system that combines apprenticeships with classroom education — the dual system. This approach brings young people into the labor force more quickly and easily. Unemployment for those in
Switzerland between the ages of 15 and 24 in 2011 was 7.7 percent; in Germany, 8.5 percent. In the United States that year, the rate was 17.3 percent, down from 18.4 percent the previous year. (A 10 percent higher rate of participation in vocational education in selected Organization for Economic
Cooperation and Development countries led to a 2 percent lower youth unemployment rate in 2011, according to economist Eric Hanushek of Stanford University.)"

Here's a bit more detail on Switzerland and on Germany:

"At ages 15 to 16, in Switzerland, about two-thirds of every cohort enter  apprenticeships, [Stefan C.] Wolter notes. Apprentices in fields from health care to hairdressing to engineering attend vocational school at least one day a week for general education and theoretical grounding for roughly three years. On other days, they apprentice under the supervision of a seasoned employee. What makes the system work so well is firm participation, which is relatively strong. “If you exclude the one-person companies and the businesses that cannot train, about 40 percent of companies that could train do train,” Wolter says. ..."

"In Germany, about 25 percent of students go to university, and apprenticeships employ another 53 percent. At 16,  they sign on for a three-year stint in one of 350 occupations. Another 15 percent may attend vocational schools. Those who are less qualified take a full-time vocational course or temporary job until they land an apprenticeship. About one-quarter of German employers participate. ..."

Other western European countries use variations of the Swiss and German model. Belgium, Finland, Sweden, and the Netherlands train most vocational students in school programs, while Germany, Switzerland, Austria, and Denmark have large school-and-work programs. The United States is an outlier: By international standards and official definitions, it has virtually no vocational education and training program."

From a U.S. perspective, it's hard to think clearly about how this kind of widespread use of apprenticeships and vocational school would even work.  Half or two-thirds of 16 year-old students involved in paid internships? A quarter or a third of all employers providing a large number of such internships as part of their regular business model? Internships across a wide array of professions, both blue- and white-collar? My American mind boggles. But given that a four-year college degree is demonstrably not a good fit for many young Americans, it's past time to take some of the alternatives more seriously.


I've posted from time to time about the merits of apprenticeships and various alternative credentials.

For examples, see this post from October 18, 2011, on "Apprenticeships for the U.S. Economy, "
this post from last November 3, 2011 on "Recognizing Non-formal and Informal Learning," and th is post from January 16, 2012 on "Certificate Programs for Labor Market Skills."

 

 

Checkerboard Puzzle, Moore's Law, and Growth Prospects

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My father the mathematician first posed the checkerboard puzzle to me back in grade-school, perhaps on some rainy Saturday. His version of the story went something like this:

The jester performs a great deed, and the king asks him how he would like to be rewarded. The jester is aware that the king is a highly volatile individual, and if the jester asks for too much, the king might just kill him then and there. The jester also knows that the king views his promise as sacred, so if the king says "yes" to the jester's proposal, then the king will honor that promise. So in a way, the jester's problem is how to ask for a lot, but have the king at least initially think it's not very much, so that the king will give his consent.

So the jester clowns around a bit and then says: "Here's all I want. Take this checkerboard. On the first square, but one piece of gold. On the second square, two pieces. On the third square, four pieces, and on the fourth square, 8 pieces. Double the amount on each square until you reach the end of the checkerboard."

In the story, the king laughs at this comic proposal and says,  "Your great deed was so wonderful, I would have happily done much more than this! I grant your request!"

But of course, when the king starts hauling up gold pieces from the treasury, he will discover that 2 raised to the 63rd power, the final spot on the checkerboard requires about 9 quintillion gold pieces (that is, 9 followed by 18 zeros). 

I've had some sense of the power of exponential growth ever since.  But what I hadn't thought about is the interaction of Moore's Law and economic growth. Moore's Law is of course named for Gordon Moore, one of the founders of Intel, who noticed this pattern back in 1965. Back in the 1970s, he wrote a paper that contained the following graph showing how much it cost to produce a computer chip with a certain number of components. Here's his figure. Notice that the numbers of component on the horizontal axis and the cost figures on the vertical axis are both graphed as logarithm (specifically, each step up the axis is a change by a factor of 10). The key takeaway was that the number of transistors ("components") on an integrated circuit was doubling about every two years, making computing power much cheaper and faster.


Ever since I started reading up on Moore's law in the early 1980s, there have been predictions in the trade press that it will soon reach technological limits and come to and end. But Moore's law marches on: indeed, the research and innovation targets at Intel and other chip-makers are defined in terms of making sure that Moore's law continues to hold for at least awhile longer. Stephen Shankland offers a nice accessible overview of the current situation in an October 15, 2012, essay on CNET: ""Moore's Law: The rule that really matters in tech"  (The Gordon Moore graph above is copied from Shankland's essay.)

As Shankland writes: "To keep up with Moore's Law, engineers must keep shrinking the size of transistors. Intel, the leader in the race, currently uses a manufacturing process with 22-nanometer features. That's 22 billionths of a meter, or roughly a 4,000th the width of a human hair." He cites a variety of industry and research experts to the effect that Moore's law has at least another decade to run--and remember, a decade of doubling every two years means five more doublings!

It's hard to wrap one's mind around what it means to say that the power of microchipo technology will increase by a factor of 32 (doubling five times) in the next 10 years. A characteristically intriguing survey essay  from the January 10 issue of the Economist on the future of innovation uses the checkerboard analogy to think about the potential effects of Moore's law. Here's a comment from the Economist essay:

Ray Kurzweil, a pioneer of computer science and a devotee of exponential technological extrapolation, likes to talk of “the second half of the chess board”. There is an old fable in which a gullible king is tricked into paying an obligation in grains of rice, one on the first square of a chessboard, two on the second, four on the third, the payment doubling with every square. Along the first row, the obligation is minuscule. With half the chessboard covered, the king is out only about 100 tonnes of rice. But a square before reaching the end of the seventh row he has laid out 500m tonnes in total—the whole world’s annual rice production. He will have to put more or less the same amount again on the next square. And there will still be a row to go.

Erik Brynjolfsson and Andrew McAfee of MIT make use of this image in their e-book “Race Against the Machine”. By the measure known as Moore’s law, the ability to get calculations out of a piece of silicon doubles every 18 months. That growth rate will not last for ever; but other aspects of computation, such as the capacity of algorithms to handle data, are also growing exponentially. When such a capacity is low, that doubling does not matter. As soon as it matters at all, though, it can quickly start to matter a lot. On the second half of the chessboard not only has the cumulative effect of innovations become large, but each new iteration of innovation delivers a technological jolt as powerful as all previous rounds combined."

Now, it's of course true that doubling the capacity of computer chips doesn't translate in a direct way into a higher standard of living: there are many steps from one to the other. But my point here is to note that many of us (myself included) have been thinking about the changes in electronics technology a little too much like the king in the checkerboard story: that is, we think of something doubling a few times, even 10 or 20 times, and we know it's a big change, but it somehow seems within our range of comprehension.

But when something has already been doubling every 18 months or two years for a half-century--it is continuing to double!--the absolute size of each additional doubling is starting to get very large. I lack the imagination to conceive of what will be done with all this cheap computing power in terms of health care, education, industrial process, communication, transportation, entertainment, food, travel, design, and more. But I suspect that these enormous repeated doublings, as Moore's law marches forward in the next decade and drives computing speeds up and prices down, will transform lives and industries in ways that we are only just starting to imagine.

Checkerboard Puzzle, Moore's Law, and Growth Prospects

To contact us Click HERE
My father the mathematician first posed the checkerboard puzzle to me back in grade-school, perhaps on some rainy Saturday. His version of the story went something like this:

The jester performs a great deed, and the king asks him how he would like to be rewarded. The jester is aware that the king is a highly volatile individual, and if the jester asks for too much, the king might just kill him then and there. The jester also knows that the king views his promise as sacred, so if the king says "yes" to the jester's proposal, then the king will honor that promise. So in a way, the jester's problem is how to ask for a lot, but have the king at least initially think it's not very much, so that the king will give his consent.

So the jester clowns around a bit and then says: "Here's all I want. Take this checkerboard. On the first square, but one piece of gold. On the second square, two pieces. On the third square, four pieces, and on the fourth square, 8 pieces. Double the amount on each square until you reach the end of the checkerboard."

In the story, the king laughs at this comic proposal and says,  "Your great deed was so wonderful, I would have happily done much more than this! I grant your request!"

But of course, when the king starts hauling up gold pieces from the treasury, he will discover that 2 raised to the 63rd power, the final spot on the checkerboard requires about 9 quintillion gold pieces (that is, 9 followed by 18 zeros). 

I've had some sense of the power of exponential growth ever since.  But what I hadn't thought about is the interaction of Moore's Law and economic growth. Moore's Law is of course named for Gordon Moore, one of the founders of Intel, who noticed this pattern back in 1965. Back in the 1970s, he wrote a paper that contained the following graph showing how much it cost to produce a computer chip with a certain number of components. Here's his figure. Notice that the numbers of component on the horizontal axis and the cost figures on the vertical axis are both graphed as logarithm (specifically, each step up the axis is a change by a factor of 10). The key takeaway was that the number of transistors ("components") on an integrated circuit was doubling about every two years, making computing power much cheaper and faster.


Ever since I started reading up on Moore's law in the early 1980s, there have been predictions in the trade press that it will soon reach technological limits and come to and end. But Moore's law marches on: indeed, the research and innovation targets at Intel and other chip-makers are defined in terms of making sure that Moore's law continues to hold for at least awhile longer. Stephen Shankland offers a nice accessible overview of the current situation in an October 15, 2012, essay on CNET: ""Moore's Law: The rule that really matters in tech"  (The Gordon Moore graph above is copied from Shankland's essay.)

As Shankland writes: "To keep up with Moore's Law, engineers must keep shrinking the size of transistors. Intel, the leader in the race, currently uses a manufacturing process with 22-nanometer features. That's 22 billionths of a meter, or roughly a 4,000th the width of a human hair." He cites a variety of industry and research experts to the effect that Moore's law has at least another decade to run--and remember, a decade of doubling every two years means five more doublings!

It's hard to wrap one's mind around what it means to say that the power of microchipo technology will increase by a factor of 32 (doubling five times) in the next 10 years. A characteristically intriguing survey essay  from the January 10 issue of the Economist on the future of innovation uses the checkerboard analogy to think about the potential effects of Moore's law. Here's a comment from the Economist essay:

Ray Kurzweil, a pioneer of computer science and a devotee of exponential technological extrapolation, likes to talk of “the second half of the chess board”. There is an old fable in which a gullible king is tricked into paying an obligation in grains of rice, one on the first square of a chessboard, two on the second, four on the third, the payment doubling with every square. Along the first row, the obligation is minuscule. With half the chessboard covered, the king is out only about 100 tonnes of rice. But a square before reaching the end of the seventh row he has laid out 500m tonnes in total—the whole world’s annual rice production. He will have to put more or less the same amount again on the next square. And there will still be a row to go.

Erik Brynjolfsson and Andrew McAfee of MIT make use of this image in their e-book “Race Against the Machine”. By the measure known as Moore’s law, the ability to get calculations out of a piece of silicon doubles every 18 months. That growth rate will not last for ever; but other aspects of computation, such as the capacity of algorithms to handle data, are also growing exponentially. When such a capacity is low, that doubling does not matter. As soon as it matters at all, though, it can quickly start to matter a lot. On the second half of the chessboard not only has the cumulative effect of innovations become large, but each new iteration of innovation delivers a technological jolt as powerful as all previous rounds combined."

Now, it's of course true that doubling the capacity of computer chips doesn't translate in a direct way into a higher standard of living: there are many steps from one to the other. But my point here is to note that many of us (myself included) have been thinking about the changes in electronics technology a little too much like the king in the checkerboard story: that is, we think of something doubling a few times, even 10 or 20 times, and we know it's a big change, but it somehow seems within our range of comprehension.

But when something has already been doubling every 18 months or two years for a half-century--it is continuing to double!--the absolute size of each additional doubling is starting to get very large. I lack the imagination to conceive of what will be done with all this cheap computing power in terms of health care, education, industrial process, communication, transportation, entertainment, food, travel, design, and more. But I suspect that these enormous repeated doublings, as Moore's law marches forward in the next decade and drives computing speeds up and prices down, will transform lives and industries in ways that we are only just starting to imagine.

Checkerboard Puzzle, Moore's Law, and Growth Prospects

To contact us Click HERE
My father the mathematician first posed the checkerboard puzzle to me back in grade-school, perhaps on some rainy Saturday. His version of the story went something like this:

The jester performs a great deed, and the king asks him how he would like to be rewarded. The jester is aware that the king is a highly volatile individual, and if the jester asks for too much, the king might just kill him then and there. The jester also knows that the king views his promise as sacred, so if the king says "yes" to the jester's proposal, then the king will honor that promise. So in a way, the jester's problem is how to ask for a lot, but have the king at least initially think it's not very much, so that the king will give his consent.

So the jester clowns around a bit and then says: "Here's all I want. Take this checkerboard. On the first square, but one piece of gold. On the second square, two pieces. On the third square, four pieces, and on the fourth square, 8 pieces. Double the amount on each square until you reach the end of the checkerboard."

In the story, the king laughs at this comic proposal and says,  "Your great deed was so wonderful, I would have happily done much more than this! I grant your request!"

But of course, when the king starts hauling up gold pieces from the treasury, he will discover that 2 raised to the 63rd power, the final spot on the checkerboard requires about 9 quintillion gold pieces (that is, 9 followed by 18 zeros). 

I've had some sense of the power of exponential growth ever since.  But what I hadn't thought about is the interaction of Moore's Law and economic growth. Moore's Law is of course named for Gordon Moore, one of the founders of Intel, who noticed this pattern back in 1965. Back in the 1970s, he wrote a paper that contained the following graph showing how much it cost to produce a computer chip with a certain number of components. Here's his figure. Notice that the numbers of component on the horizontal axis and the cost figures on the vertical axis are both graphed as logarithm (specifically, each step up the axis is a change by a factor of 10). The key takeaway was that the number of transistors ("components") on an integrated circuit was doubling about every two years, making computing power much cheaper and faster.


Ever since I started reading up on Moore's law in the early 1980s, there have been predictions in the trade press that it will soon reach technological limits and come to and end. But Moore's law marches on: indeed, the research and innovation targets at Intel and other chip-makers are defined in terms of making sure that Moore's law continues to hold for at least awhile longer. Stephen Shankland offers a nice accessible overview of the current situation in an October 15, 2012, essay on CNET: ""Moore's Law: The rule that really matters in tech"  (The Gordon Moore graph above is copied from Shankland's essay.)

As Shankland writes: "To keep up with Moore's Law, engineers must keep shrinking the size of transistors. Intel, the leader in the race, currently uses a manufacturing process with 22-nanometer features. That's 22 billionths of a meter, or roughly a 4,000th the width of a human hair." He cites a variety of industry and research experts to the effect that Moore's law has at least another decade to run--and remember, a decade of doubling every two years means five more doublings!

It's hard to wrap one's mind around what it means to say that the power of microchipo technology will increase by a factor of 32 (doubling five times) in the next 10 years. A characteristically intriguing survey essay  from the January 10 issue of the Economist on the future of innovation uses the checkerboard analogy to think about the potential effects of Moore's law. Here's a comment from the Economist essay:

Ray Kurzweil, a pioneer of computer science and a devotee of exponential technological extrapolation, likes to talk of “the second half of the chess board”. There is an old fable in which a gullible king is tricked into paying an obligation in grains of rice, one on the first square of a chessboard, two on the second, four on the third, the payment doubling with every square. Along the first row, the obligation is minuscule. With half the chessboard covered, the king is out only about 100 tonnes of rice. But a square before reaching the end of the seventh row he has laid out 500m tonnes in total—the whole world’s annual rice production. He will have to put more or less the same amount again on the next square. And there will still be a row to go.

Erik Brynjolfsson and Andrew McAfee of MIT make use of this image in their e-book “Race Against the Machine”. By the measure known as Moore’s law, the ability to get calculations out of a piece of silicon doubles every 18 months. That growth rate will not last for ever; but other aspects of computation, such as the capacity of algorithms to handle data, are also growing exponentially. When such a capacity is low, that doubling does not matter. As soon as it matters at all, though, it can quickly start to matter a lot. On the second half of the chessboard not only has the cumulative effect of innovations become large, but each new iteration of innovation delivers a technological jolt as powerful as all previous rounds combined."

Now, it's of course true that doubling the capacity of computer chips doesn't translate in a direct way into a higher standard of living: there are many steps from one to the other. But my point here is to note that many of us (myself included) have been thinking about the changes in electronics technology a little too much like the king in the checkerboard story: that is, we think of something doubling a few times, even 10 or 20 times, and we know it's a big change, but it somehow seems within our range of comprehension.

But when something has already been doubling every 18 months or two years for a half-century--it is continuing to double!--the absolute size of each additional doubling is starting to get very large. I lack the imagination to conceive of what will be done with all this cheap computing power in terms of health care, education, industrial process, communication, transportation, entertainment, food, travel, design, and more. But I suspect that these enormous repeated doublings, as Moore's law marches forward in the next decade and drives computing speeds up and prices down, will transform lives and industries in ways that we are only just starting to imagine.

Obesity and Healthy Snacks

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The rise in American rates of obesity can be traced back to what seems like a fairly small rise in daily calories consumed, I learned this lesson from an article on the causes of obesity about 10 years back in my own Journal of Economic Perspectives. In "Why Have Americans Become More Obese?" David M. Cutler, Edward L. Glaeser and Jesse M. Shapiro wrote that the "10- to 12-pound increase in median weight we observe in the past two decades requires a net caloric imbalance of about 100 to 150 calories per day. These calorie numbers are strikingly small. One hundred and fifty calories per day is three Oreo cookies or one can of Pepsi. It is about a mile and a half of walking."

Elizabeth Frazao, Hayden Stewart, Jeffrey Hyman, and Andrea Carlson apply a similar logic in "Gobbling Up Snacks: Cause or Potential Cure for Childhood Obesity?" which appears in the n the December 2012 issue of Amber Waves, published by the Economic Research Service at the U.S. Department of Agriculture. 

When I was a child, my mother had a clear-cut policy on before-dinner snacks: I was allowed to eat all the raw carrots I wanted. The parental philosophy appears to have gone out of date. The USDA economists explain: "Consumption of snacksamong children has increased markedly over the last 35 years. Inthe late 1970s, American children consumed an average of only onesnack a day. Today, they are consuming nearly three snacks per day.As a result, daily calories from children's snacks have increasedby almost 200 calories over the period." They propose that a shift to healthier snacks could play a useful role in cutting childhood obesity.

As a starting point, here's a table showing the calorie count for some fruit and vegetable stacks, compared with some commonly consumed snack foods. As a parent of three, I confess that I can't see my children snacking on broccoli florets. But grapes, strawberries, cantaloupe, or apples are all possibilities, and even among the snack foods, some choices like popsicles or fruit rolls are at least better than the alternatives.

One common reaction to this sort of list is that the healthy snacks cost too much. It is true that cookies, crackers, and chips make a relatively cheap snack, and some of the healthier choices, like tangerines, grape tomatoes, and strawberries are a costlier snack. But some of the unhealthy choices like muffins and Danish are also on the costly side, while bananas, oranges, and those good old carrot sticks are not especially pricey. Here are their calculations of cost-per-snack.

 Generalizing wildly from personal experience, as social scientists are wont to do, it feels to me as if a cultural shift has occurred about a "suitable" snack. Two of my children were, at different times, in soccer leagues where the parents organized themselves to make sure that each child got a "treat" of a cookie or chips and a large sugared drink after each game. When I coached a different youth soccer team, I brought bags of orange and apple slices for the players. As far as I could tell, the kids were equally happy with the oranges--at least once it was clear that cookies and chips would not be forthcoming.

Consuming healthier (and fewer) snacks could make a real difference to child obesity--and it makes sense for adults, too.

Checkerboard Puzzle, Moore's Law, and Growth Prospects

To contact us Click HERE
My father the mathematician first posed the checkerboard puzzle to me back in grade-school, perhaps on some rainy Saturday. His version of the story went something like this:

The jester performs a great deed, and the king asks him how he would like to be rewarded. The jester is aware that the king is a highly volatile individual, and if the jester asks for too much, the king might just kill him then and there. The jester also knows that the king views his promise as sacred, so if the king says "yes" to the jester's proposal, then the king will honor that promise. So in a way, the jester's problem is how to ask for a lot, but have the king at least initially think it's not very much, so that the king will give his consent.

So the jester clowns around a bit and then says: "Here's all I want. Take this checkerboard. On the first square, but one piece of gold. On the second square, two pieces. On the third square, four pieces, and on the fourth square, 8 pieces. Double the amount on each square until you reach the end of the checkerboard."

In the story, the king laughs at this comic proposal and says,  "Your great deed was so wonderful, I would have happily done much more than this! I grant your request!"

But of course, when the king starts hauling up gold pieces from the treasury, he will discover that 2 raised to the 63rd power, the final spot on the checkerboard requires about 9 quintillion gold pieces (that is, 9 followed by 18 zeros). 

I've had some sense of the power of exponential growth ever since.  But what I hadn't thought about is the interaction of Moore's Law and economic growth. Moore's Law is of course named for Gordon Moore, one of the founders of Intel, who noticed this pattern back in 1965. Back in the 1970s, he wrote a paper that contained the following graph showing how much it cost to produce a computer chip with a certain number of components. Here's his figure. Notice that the numbers of component on the horizontal axis and the cost figures on the vertical axis are both graphed as logarithm (specifically, each step up the axis is a change by a factor of 10). The key takeaway was that the number of transistors ("components") on an integrated circuit was doubling about every two years, making computing power much cheaper and faster.


Ever since I started reading up on Moore's law in the early 1980s, there have been predictions in the trade press that it will soon reach technological limits and come to and end. But Moore's law marches on: indeed, the research and innovation targets at Intel and other chip-makers are defined in terms of making sure that Moore's law continues to hold for at least awhile longer. Stephen Shankland offers a nice accessible overview of the current situation in an October 15, 2012, essay on CNET: ""Moore's Law: The rule that really matters in tech"  (The Gordon Moore graph above is copied from Shankland's essay.)

As Shankland writes: "To keep up with Moore's Law, engineers must keep shrinking the size of transistors. Intel, the leader in the race, currently uses a manufacturing process with 22-nanometer features. That's 22 billionths of a meter, or roughly a 4,000th the width of a human hair." He cites a variety of industry and research experts to the effect that Moore's law has at least another decade to run--and remember, a decade of doubling every two years means five more doublings!

It's hard to wrap one's mind around what it means to say that the power of microchipo technology will increase by a factor of 32 (doubling five times) in the next 10 years. A characteristically intriguing survey essay  from the January 10 issue of the Economist on the future of innovation uses the checkerboard analogy to think about the potential effects of Moore's law. Here's a comment from the Economist essay:

Ray Kurzweil, a pioneer of computer science and a devotee of exponential technological extrapolation, likes to talk of “the second half of the chess board”. There is an old fable in which a gullible king is tricked into paying an obligation in grains of rice, one on the first square of a chessboard, two on the second, four on the third, the payment doubling with every square. Along the first row, the obligation is minuscule. With half the chessboard covered, the king is out only about 100 tonnes of rice. But a square before reaching the end of the seventh row he has laid out 500m tonnes in total—the whole world’s annual rice production. He will have to put more or less the same amount again on the next square. And there will still be a row to go.

Erik Brynjolfsson and Andrew McAfee of MIT make use of this image in their e-book “Race Against the Machine”. By the measure known as Moore’s law, the ability to get calculations out of a piece of silicon doubles every 18 months. That growth rate will not last for ever; but other aspects of computation, such as the capacity of algorithms to handle data, are also growing exponentially. When such a capacity is low, that doubling does not matter. As soon as it matters at all, though, it can quickly start to matter a lot. On the second half of the chessboard not only has the cumulative effect of innovations become large, but each new iteration of innovation delivers a technological jolt as powerful as all previous rounds combined."

Now, it's of course true that doubling the capacity of computer chips doesn't translate in a direct way into a higher standard of living: there are many steps from one to the other. But my point here is to note that many of us (myself included) have been thinking about the changes in electronics technology a little too much like the king in the checkerboard story: that is, we think of something doubling a few times, even 10 or 20 times, and we know it's a big change, but it somehow seems within our range of comprehension.

But when something has already been doubling every 18 months or two years for a half-century--it is continuing to double!--the absolute size of each additional doubling is starting to get very large. I lack the imagination to conceive of what will be done with all this cheap computing power in terms of health care, education, industrial process, communication, transportation, entertainment, food, travel, design, and more. But I suspect that these enormous repeated doublings, as Moore's law marches forward in the next decade and drives computing speeds up and prices down, will transform lives and industries in ways that we are only just starting to imagine.