In The Souls of Black Folk, what is the "Negro problem" and why can't Du Bois answer it?

The "Negro problem" is characterized by Du Bois as a "veil" that separates black and white consciousness. Black citizens necessarily develop a "double consciousness" in US society in which their own self-perceptions are at war with the way they are perceived by white society.

African Americans therefore become a problem to white citizens because these white citizens refuse to fully accept them as equal human beings, instead marginalizing them as "others." This makes it difficult for black citizens to integrate into American life and a struggle for them to develop a positive self image and strong sense of self. African Americans, Du Bois contends, live double lives, presenting one image—most often the expected image—of who they are to white citizens while living and feeling a very different way.

At the beginning of The Souls of Black Folk, Du Bois mentions being treated as a "problem" for being black. He says that in response to the implied question whites often dance around, he seldom answers:

To the real question, How does it feel to be a problem? I answer seldom a word.

A little later in the introduction, however, he does try to answer the question, by saying it is painful to be considered a problem. This othering makes it difficult for African Americans to find a true sense of identity and worth. He writes:

the Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world,—a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity.

Du Bois spends much of his book arguing that the way forward for black citizens is not to accommodate themselves to second class citizenship but to fight for full civil rights and equality with white citizens.

In The Souls of Black Folk, what is the "Negro problem" and why can't Du Bois answer it?

The negro problem, as it was called, was the issue of what the place of African Americans in society should be.  They were no longer slaves, but the vast majority of white people did not consider them to be equal.  So what place was there for them between slaves and equals?  As James Baldwin once said:

At the root of the American Negro problem is the necessity of the American white man to find a way of living with the Negro...

I'm assuming that you're talking about the very beginning of the book where he says "How does it feel to be a problem.  I answer seldom a word."

I believe he says this because he resents being seen as a problem and not as a person.  He resents the way that people try to pretend that they accept black people ("I know an excellent colored man in town.") even when they really see him as a problem.

How does Du Bois use the word problem in The Souls of Black Folk?

Du Bois uses the word problem frequently in this text to mean racism, which he calls the problem of the color line.

Writing in 1900, though his words are as relevant 120 years later as they were then, Du Bois understands racism as the major problem ripping apart the American soul. He states,

the problem of the Twentieth Century is the problem of the color line [racism].

He spends the rest of the book exploring and describing this problem from a Black perspective.

The problem with racism is not merely abstract: Blacks as individuals are treated as a "problem" by the dominant white society. He states that it is a strange feeling to be treated as a problem, not a person, just for having black skin.

Du Bois argues that this problem can only be solved by Black people achieving full equality with white people and being accepted as equals so that the color line will disappear. This seems obvious to us now, but it is important to understand the context in which Du Bois writes.

At that point, an accommodationist approach to race, championed by Black leader Booker T. Washington, was dominant. It argued that Black people should concentrate on building up an economic base and not worry about equal rights until later. Du Bois asserts that Black problems, including economic poverty, cannot be solved until this fundamental problem of the "color line" is addressed head on and overcome.

What does Du Bois mean by "problem" in the context of the "color-line" and in reference to himself in The Souls of Black Folk?

There is a difference between the personal sense or perception of being a "problem," as expressed in Chapter 1: Of Our Spiritual Strivings, and the "problem of the color-line," as mentioned in Chapter 2: Of the Dawn of Freedom. However, the former condition was precipitated by the latter. DuBois's sense of being a "problem"—a notion that is imposed on him by the liberal, well-meaning whites he alludes to in his prose—is a result of the history that he describes in the second chapter, which constructed black people, first, as non-citizens and "three-fifths" of a person, then, a system that imposed a second-class status and eliminated due process.

Being unwanted, being a "problem," can result in a feeling of defeat or, in the case of DuBois, in a determination to strive: "That sky was bluest when I could beat my mates at examination-time, or beat them at a foot-race, or even beat their stringy heads." Still, the historical framework that constructed him as a "problem" causes that striving to be for naught in many instances: "Alas, with the years all this fine contempt began to fade; for the worlds I longed for, and all their dazzling opportunities, were theirs, not mine."

The perception of DuBois and other black people as a "problem" prevents whites, based on his anecdotes, from speaking to black people as individuals. Instead, black people are spoken to within the context of their racial oppression, from the understanding of their position as members of a permanent underclass. Thus, the "problem" of one's existence as a black person is created by the conditions of history and reinforced even by those who mean well.

What does Du Bois mean by "problem" in the context of the "color-line" and in reference to himself in The Souls of Black Folk?

In the first instance, Du Bois uses "problem" to describe a generic, abstract, widespread issue: the issue of the divide between blacks and whites (what he calls the color-line) caused by racism. Race, as an abstract concept children learn, causes white people to believe themselves superior to black people. It also causes black people to internalize the cultural perception of inferiority in such a way that they display two faces to the world: the servile face they turn to whites and the reality of their own experience, understood only by other black people. This first instance is "problem" used in a very broad way.

In the second instance, Du Bois takes the broad concept of a "color-line" problem and makes it individual and specific. On the big, generic level, this "problem" is nothing more than a false construct: in reality, there should be no "color-line" because black and white people have equal ability, but the imposition of racism has created a false reality. Nevertheless, this false construct has very real consequences when it becomes a "problem" faced by a living, individual black person like Du Bois, who is held in contempt and treated as lesser—and ultimately has many fewer opportunities—based on a larger, more abstract perception of a "problem."

Ideology, in other words, matters. If white people believe black people are problem, then they become a problem.

What does DuBois identify as the 20th-century problem and what does "the color-line" mean?

In W.E.B. DuBois's seminal work, The Souls of Black Folk, the problem of the twentieth-century is, indeed, that of the "color-line."

The phrase "the color-line" first appeared in an article called "The Color Line" written by Frederick Douglass for the North American Review in 1881. It became better-known when DuBois used the phrase to expand its relevance for the issues confronting black Americans in the twentieth-century.

The color-line is a direct reference to the segregation that existed in the United States -- both the legal, de jure, segregation in the South, and the de facto, or "in effect," racism in the North and the West. Segregation made it so that blacks and whites lived in such disparate worlds that their lives generally did not look at all the same. Though black people were very often aware the rights and privileges that were kept from them, particularly since working-class blacks were very often employed in white homes as domestic laborers, many white people were blind -- often willfully -- to the inferior conditions in which black people existed.

The color-line created inequalities in education, housing, and access to employment. It made it nearly impossible for black and white people to have any amity between them. Existentially, it worked to convince black people of their presumed inferiority. 

However, DuBois was concerned not only with the conditions in which black people existed in the United States, but also the lives of darker peoples in other countries, particularly the colonies in Africa, Asia, and Oceania. The subjugation of these peoples was justified by pseudo-scientific racial theories which attempted to give "the color-line" a scientific basis.