Opinion for the Court filed by Circuit Judge MIKVA.
MIKVA, Circuit Judge:
Appellants were convicted of a wide variety of criminal charges after thirteen weeks of trial and eight weeks of jury deliberations. We find that the district court’s dismissal of a juror in the midst of the jury’s lengthy deliberations deprived the appellants of their constitutional right to a unanimous jury. We must therefore reverse the convictions.
I. Background
All nine of the appellants in this case are members of a religious group formally titled the “Original Hebrew Israelite Nation of the Kingdom of Jerusalem” and commonly called the “Nation.” Members of the Nation are organized into congregations called “missions”; astride all of these missions are several “ministries” or governing bodies. Appellant Warren Brown is the head of one of the Nation’s ministries, which is located in Chicago; appellants James Stone and Gerald Bethea are members of Brown’s staff. Appellant J.C. Vor-tis leads the Nation’s Baltimore/Washington-area Mission. The remaining appellants in this case — Gregory Coles, Thomas Cavin, Cordell Debardelaben, Daryl Gris-som, and Kenneth Robinson — belong to the Baltimore/Washington-area Mission.
In September of 1985, a grand jury returned a sixty-nine-count indictment against the nine appellants. Count 1 of the indictment charged each appellant with conspiring to conduct the affairs of an enterprise through a pattern of racketeering activity, in violation of section 1962(d) of the Racketeer Influenced and Corrupt Organizations Act (RICO), 18 U.S.C. § 1961 et seq. (1984). Count 2 of the indictment charged each appellant with actually conducting the affairs of an enterprise through a pattern of racketeering activity, in violation of section 1962(c) of RICO. The enterprise alleged in these two counts was “a group of individuals associated in fact which operated within the framework of a larger group ... [called] The Nation” and organized itself along similar hierarchical lines. The pattern of racketeering activity alleged in the two counts consisted of fifteen discrete acts, each of which fell within section 1961(l)’s definition of “racketeering activity.” Subsequent counts of the indictment listed these fifteen predicate acts as separate and independent offenses. Still other counts charged offenses that were not also listed as RICO predicate acts because they did not fall within section 1961(l)’s definition of “racketeering activity.”
The multitude of offenses charged in the indictment arose from a smaller number of alleged criminal schemes. The most common kind of scheme alleged was the so-called “shopping spree.” According to the indictment, in each of these sprees, a member of the enterprise, employing an assumed name and counterfeit identification, opened a checking account with a small cash deposit at a local bank. The member then deposited forged checks into the account, thereby artificially inflating its balance. A short time later, the member went shopping. In essence, the member wrote worthless checks as payment for valuable goods, using a counterfeit driving license and counterfeit credit cards as supporting identification. (Occasionally, the member simply used the counterfeit credit cards to pay for the desired goods.) When the bank discovered that the original deposit checks were forgeries and refused to honor a written check, the member abandoned the account. Another common kind of criminal scheme described in the indictment was the theft and use of blank airline ticket stock. According to the indictment, in each scheme of this kind, a member of the enterprise stole a large quantity of blank ticket stock from an airline company. Members of the enterprise then prepared counterfeit tickets from the stock and either used them for personal travel or sold them to individuals not associated with the enterprise. Finally, the indictment alleged that members of the enterprise had participated in two schemes to defraud the government. In one scheme, members of the enterprise allegedly obtained welfare benefits by fraud; in the other scheme, members allegedly planned to obtain social security benefits in a fraudulent manner.
Trial of the nine appellants commenced on March 10, 1986 and continued for thirteen weeks. In the government’s case-in-chief, prosecutors presented testimony from more than one hundred witnesses and introduced a mass of documentary and physical evidence, including handwriting analyses, fingerprint identifications, and stolen credit cards and checks. The prosecutors also played to the jury nearly one hundred hours of tape-recorded telephone conversations between members of the alleged enterprise. The appellants earlier had objected to the admission of these recordings on the ground that they had been illegally obtained, but the trial judge had refused to suppress the evidence.
Six of the nine appellants represented themselves for large portions of the trial. On March 24, 1986, after two weeks of trial, Stone, Bethea, Vortis, Cavin, and Robinson dismissed their court-appointed counsel; each of these appellants represented himself for the remainder of the trial. On May 27, the fourth day of the defense case, Brown discharged his retained counsel; he too represented himself for the duration of the trial. Prior to allowing any of these appellants to proceed pro se, the court questioned the voluntariness of their decisions and warned them of the hazards of self-representation present in all serious criminal cases.
On June 6, the jury began its deliberations. On July 8, after five weeks of deliberations, the jury sent a note to the judge asking: “When is a defendant not guilty? When all jurors give a unanimous verdict vote of not guilty or, at least, one gives a vote of not guilty?” VI Joint Appendix (J.A.) at 7246. The next morning, the court returned a note to the jury stating:
In response to your note of yesterday, I instruct you as follows: In order to return a verdict of either guilty or not guilty as to a defendant on any count, your decision must be unanimous. With respect to those counts and those defendants on which you have not yet reached agreement, please continue your deliberations in an effort to reach an unanimous verdict of either guilty or not guilty.
Id. at 7252. On the afternoon of the same day, one of the jurors sent a note to the court stating: “I Bernard Spriggs am not able to discharge my duties as a member of this jury.” Id. at 7257. After conferring with counsel, the court decided to call Spriggs into the courtroom and attempt to discover his reason for wanting to quit deliberations. When Spriggs entered the courtroom, the following discussion, quoted in its entirety, occurred:
COURT: I have your note which reads, “I Bernard Spriggs, am not able to discharge my duties as a member of this jury.” That is your note, is it?
SPRIGGS: Yes, it’s my note.
COURT: May I ask you this question: Does this have to do with your health?
SPRIGGS: No.
COURT: Up to this point have you been able to, as you say, discharge your duties?
SPRIGGS: Up to this point, yes.
COURT: Now, I don’t want to know how you have voted, or the jury has voted, on anything with respect to any defendant.
But can you tell us just generally what the nature of the problem is. Could it be a personality problem between you and other members, or any one or more members of the jury, or is it something else?
SPRIGGS: No; it’s not a personality problem. It’s the way the R.I.C.O. conspiracy act reads. And considering how it runs, I cannot—
COURT: All right.
You cannot what?
SPRIGGS: I cannot—
COURT: Do you understand it?
SPRIGGS: I understand it.
COURT: I see.
SPRIGGS: But at this point I can’t go along with that act.
If I had known at the beginning of this trial what the act said, I would have not said I could be impartial.
COURT: Because what you’re saying is you don’t like the law and you can’t follow the law and my instructions on the law because you disagree with it; is that what you mean?
SPRIGGS: I disagree with it; yes.
COURT: And, is that what you mean when you say you’re not able to discharge your duties, because you can’t follow the law? You disagree with the law? And if you had known more clearly what the law said—
SPRIGGS: Yes.
COURT: —You would have indicated on the voir dire—
SPRIGGS: Yes, sir; Yes, sir.
COURT: —When I asked you that question, would you follow the court’s instructions, you would say “No, because I don’t like the law”?
SPRIGGS: It’s the way it’s written and the way the evidence has been presented.
COURT: If the law were different, maybe you could go along — but your understanding—
SPRIGGS: If the evidence was presented in a fashion in which the law is written, then, maybe, I would be able to discharge my duties. But to me—
COURT: I don’t want to know anything about your individual verdict or expression. I just want to know — we were trying to find out what the nature of your problem was because you ve been on this case since the third of March and you’ve [sic] haven’t missed a beat.
SPRIGGS: No.
COURT: What I’d ask you to do-Ralph, would you ask — Mr. Spriggs, would you just go and sit in my chambers for a minute and let me have an opportunity to talk to the lawyers and then I’ll get back to you. Thank you.
Id. at 7265-67. The court discussed the issue with counsel that afternoon and the next morning. The prosecution argued that Spriggs had stated that he would not follow the law and further contended that the court had a duty to dismiss a juror who would not follow the law, even after the commencement of deliberations. Defense counsel argued that Spriggs had expressed difficulties with the evidence and that dismissing him in these circumstances would violate the defendants’ constitutional rights. The court at one point expressed the view that Spriggs’ reasons for thinking himself incapable of performing his functions as a juror were not entirely clear. The judge decided, however, that he could not question Spriggs any further because additional inquiry would intrude on the secrecy of the jury’s deliberations. The judge then determined to dismiss Spriggs under Federal Rule of Criminal Procedure 23(b), which allows a court to excuse a juror if “necessary” and “for just cause” after the start of deliberations, on the ground that Spriggs would not follow the law and thus could not discharge his duty as a juror. The court admonished the remaining eleven jurors that the dismissal was not to influence their consideration of the case and instructed them to return to deliberations.
The jury returned its verdict on July 30, eight weeks after the start of deliberations and three weeks after the discharge of Spriggs. The jury convicted the appellants of both of the RICO violations alleged in the indictment and also returned convictions on most of the other offenses charged. The court entered a judgment on the verdict and two weeks later sentenced the appellants to lengthy prison terms.
The appellants, who already have begun to serve these prison terms, now appeal the court’s entry of judgment on the jury verdict. The appellants present a wide variety of arguments in support of reversing some or all of the convictions. Most notably, the appellants urge that we must reverse the convictions because the district court’s dismissal of juror Spriggs deprived them of their constitutional right to a unanimous jury. The appellants also present arguments relating to the joinder of offenses in the indictment, the court’s response to the requests of several appellants to proceed 'pro se, and the court’s admission into evidence of the tape-recorded telephone conversations.
II. Discussion
A. Dismissal of Juror Spriggs
The Sixth Amendment to the Constitution provides that “[i]n all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed.” In Apodaca v. Oregon, 406 U.S. 404, 92 S.Ct. 1628, 32 L.Ed.2d 184 (1972), five members of the Supreme Court interpreted this amendment to endow a federal criminal defendant with the right to a unanimous verdict. See 406 U.S. at 369-71, 92 S.Ct. at 1637 (Powell, J., concurring); id. at 382, 92 S.Ct. at 1644 (Douglas, J., dissenting); id. at 395, 92 S.Ct. at 1650 (Brennan, J., dissenting); id. at 400, 92 S.Ct. at 1652 (Marshall, J., dissenting); id. at 414, 92 S.Ct. at 1634 (Stewart, J., dissenting). This circuit has reiterated the Justices’ constitutional interpretation. See United States v. Essex, 734 F.2d 832, 840-41 (D.C.Cir.1984). In addition, the Federal Rules of Criminal Procedure echo the constitutional requirement of unanimity: Rule 31(a) states simply that “[t]he verdict shall be unanimous.”
The appellants claim that the district court’s decision to dismiss juror Spriggs from the jury violated their right to a unanimous verdict. They first aver that Spriggs requested dismissal not because he had principled objections to the RICO statute, but because he had doubts about the sufficiency of the government’s evidence. According to the appellants, the dismissal of a juror for this reason violates a defendant’s right to a unanimous verdict because the dismissal allows the prosecutor to gain a conviction even though he has failed to persuade all of the jurors that the defendant violated the law. The appellants next argue that even if Spriggs’ discomfort with the case stemmed solely from a principled objection to applying the substantive law, the court could not constitutionally discharge him from the jury. In essence, the appellants contend that a juror will vote to convict a defendant only when the juror believes both that the defendant violated the law and that the jury should give effect to the law. In the appellants’ view, the discharge of a juror for refusing to follow the law violates the defendant’s right to a unanimous verdict because the discharge allows a conviction to occur even though not all members of the jury wish to apply the law to the defendant’s conduct. The appellants recognize, in making these arguments, that Rule 23(b) of the Federal Rules of Criminal Procedure authorizes the discharge of a juror after deliberations have commenced “if ... necessary” and “for just cause.” The appellants aver, however, that use of the Rule to discharge a juror either because he has doubts about the sufficiency of the government’s evidence or because he refuses to apply the substantive law violates a defendant’s constitutional right to a unanimous jury.
We agree with the appellants that a court may not dismiss a juror during deliberations if the request for discharge stems from doubts the juror harbors about the sufficiency of the government’s evidence. Indeed, we think that this part of the appellants’ argument is scarcely debatable. If a court could discharge a juror on the basis of such a request, then the right to a unanimous verdict would be illusory. A discharge of this kind would enable the government to obtain a conviction even though a member of the jury that began deliberations thought that the government had failed to prove its case. Such a result is unacceptable under the Constitution. Thus, when a request for dismissal stems from the juror’s view of the sufficiency of the evidence that the government offered at trial, a judge may not discharge the juror: the judge must either declare a mistrial or send the juror back to deliberations with instructions that the jury continue to attempt to reach agreement.
We must, however, confront the problem that the reasons underlying a request for a dismissal will often be unclear. As the district court in this case aptly noted, a court may not delve deeply into a juror’s motivations because it may not intrude on the secrecy of the jury’s deliberations. Thus, unless the initial request for dismissal is transparent, the court will likely prove unable to establish conclusively the reasons underlying it. Given these circumstances, we must hold that if the record evidence discloses any possibility that the request to discharge stems from the juror’s view of the sufficiency of the government’s evidence, the court must deny the request. Cf. Essex, 734 F.2d at 843 (finding a violation of the defendant's right to a unanimous verdict because there was a “possibilit[y]” that the absent juror was “a lone holdout for innocence in the face of a hostile pro-conviction majority” and that the prosecution had not carried its “burden of convincing the entire membership of the jury”); United States v. Stratton, 779 F.2d 820, 832 (2d Cir.1985) (holding that the discharge of a juror to observe a religious holiday was proper when “the record d[id] not present even the slightest basis to believe that Juror No. 10 [posed] an obstacle to reaching a unanimous verdict”), cert. denied, - U.S.-, 106 S.Ct. 2285, 90 L.Ed.2d 726 (1986). Any other holding would fail to protect adequately a defendant’s right to be convicted only by a unanimous jury.
The record evidence in this case indicates a substantial possibility that juror Spriggs requested to be discharged because he believed that the evidence offered at trial was inadequate to support a conviction. At the beginning of his colloquy with the judge, Spriggs stated that his difficulties with the case arose from the terms of the RICO statute-"what the act said" and "how it runs." As the judge continued to question Spriggs, however, the juror began to speak of the evidence offered at the trial. Spriggs specifically stated that his difficulty was with "the way [the act is] written and the way the evidence has been presented" (emphasis added); he further noted that "[i]f the evidence was presented in a fashion in which the law is written, then, maybe, I would be able to discharge my duties." These statements, at the very least, create an ambiguous record. The district court itself recognized as much. See VI J.A. at 7297-98, 7309, 7312, 7314. We may not be able to say for a certainty that Spriggs' desire to leave the jury stemmed from his view of the adequacy of the government's evidence. But we cannot say with any conviction that Spriggs' request to be dismissed stemmed from something other than this view. Given the pos. sibility-which in this case we think a likelihood-that Spriggs' desire to quit deliberations stemmed from his belief that the evidence was inadequate to support a conviction, we must find that his dismissal violated the appellants' right to a unanimous jury verdict.
We note in conclusion, and in response to the government's primary argument on appeal, that the terms of Rule 23(b) in no way undermine our approach. As we have noted, Rule 23(b), under which the district court dismissed juror Spriggs, authorizes the discharge of a juror after deliberations have commenced "if ... necessary" and "for just cause." The government argues that our holding will "effectively read 23(b) out of the federal rules-[the Rule's] purpose is precisely to make available a remedy between the extremes of mistrial and doing nothing." Brief for Appellee at 87. This assertion, however, is invalid. First, to the extent that we limit the ability of courts to use Rule 23(b), we do so by command of the Constitution. When, as in this case, the Constitution precludes a particular application of a rule or statute, that is the end of the matter. Second, our holding cannot reasonably be said to erase Rule 23(b) from the federal books. Courts may use Rule 23(b) in many circumstances to discharge a juror. For example, other appellate courts have held that a district court may properly use Rule 23(b) to discharge a juror who has suffered serious injury, see United States v. Smith, 789 F.2d 196, 204-05 (3d Cir.), cert. denied, - U.S. -, 107 S.Ct. 668, 93 L.Ed.2d 720 (1986), who has viewed relevant materials not in evidence, see United States v. Gambino, 788 F.2d 938, 946-49 (3d Cir.), cert. denied, - U.S. -, 107 S.Ct. 98, 93 L.Ed.2d 49 (1986), and who has expressed a desire to observe a lengthy religious holiday, see Stratton, 779 F.2d at 831-32. These holdings are fully consistent with our own given that in none of these circumstances is there any reason to believe that the government has failed to persuade all of the jurors who began deliberations of the adequacy of the government's case. In addition, we specifically leave open the question that appellants raise of whether a court may constitutionally apply Rule 23(b) to discharge a juror for refusing to apply the relevant substantive law. We today hold only that Rule 23(b) is not available when the record evidence discloses a possibility that the juror believes that the government has failed to present sufficient evidence to support a conviction. Because the record evidence in this case discloses just such a possibility, we must reverse the convictions.
B. Guidance to District Court on Retrial
Our decision to reverse the convictions in this case because of the discharge of juror Spriggs may well lead to a retrial of the appellants. In a retrial, the district court will almost certainly confront four of the many issues that the appellants have raised on appeal. We have received the benefit of full briefing and argument on these four issues, and we believe it appropriate to discuss them briefly. We hope these preca-tory words will promote an equitable, orderly, and efficient retrial, if a retrial indeed takes place.
The appellants have argued to this court that the government's decision to join the offenses that were not chargeable as RICO predicate acts to the other offenses in the indictment violated Federal Rule of Criminal Procedure 8(b), which controls joinder of offenses in indictments that charge two or more defendants. The appellants appear to take the view that offenses not chargeable as RICO predicate acts may never be joined to an indictment alleging RICO offenses and offenses chargeable as RICO predicate acts. We presume that the appellants will press this argument below in an attempt to sever some of the offenses. The argument, however, has no merit. Rule 8(b) allows the joinder of offenses when they arise out of a single “series of acts or transactions.” This court has stated a “series of acts or transactions” is “two or more acts or transactions connected together or constituting parts of a common scheme or plan.” United States v. Perry, 731 F.2d 985, 990 (D.C.Cir.1984). Thus, an offense that is chargeable as a RICO predicate may be joined to an offense that is not chargeable as a RICO predicate so long as the two offenses satisfy this test of “logical relationship.” Id. We think that all of the offenses charged in this indictment are related in such a way as to make joinder appropriate. The offenses that were not chargeable as RICO predicates were committed at the same times, by the same persons, in accordance with the same general methods, and in pursuance of the same broad schemes as the offenses chargeable as RICO predicates. All of the offenses charged, in other words, arose out of the same “series of acts or transactions”; the offenses were therefore properly joined in a single indictment.
The appellants also have contended that the district court erred in admitting the tape-recorded telephone conversations into evidence at the trial. The government garnered these recordings by placing wiretaps on four telephones in the District of Columbia area. A district court judge had authorized these wiretaps on the basis of an FBI agent’s affidavit. Under the federal wiretap-authorization statute, a judge may authorize a wiretap only if the government has made a particularized showing that traditional “investigative procedures have been tried and failed or ... reasonably appear to be unlikely to succeed if tried or to be too dangerous.” 18 U.S.C. § 2518(l)(c), (3)(c) (1970). The appellants concede that the government made such a showing with respect to the “Chicago defendants” (.e., Brown, Stone, and Bethea); traditional investigative techniques had failed to uncover these defendants’ activities. The appellants aver, however, that the government failed to make such a showing with respect to the “Washington-area defendants.” In these circumstances, appellants claim, the court reviewing the affidavit could lawfully have authorized the government to tap the phones of the Chicago defendants, but had no authority to permit the government to tap the phones of the Washington-area defendants. The appellants conclude that the trial court should therefore have suppressed the evidence derived from the four wiretaps.
We believe that this argument misconstrues both the government’s affidavit and the relevant case law. The affidavit did indicate that traditional investigative techniques had yielded some evidence against the Washington-area defendants. But the affidavit also showed that these techniques had failed — and would likely continue to fail — to disclose the full nature and extent of the conspiracy of which the Washington-area defendants were a part. This circuit already has held that a court may authorize the wiretap of the phone of a member of an operation if traditional investigative techniques have proved inadequate to reveal the operation’s full “nature and scope.” United States v. Williams, 580 F.2d 578, 590 (D.C.Cir.), cert. denied, 439 U.S. 832, 99 S.Ct. 112, 58 L.Ed.2d 127 (1978); see id. at 590 n. 73; see also United States v. Vento, 533 F.2d 838, 850 (3d Cir.1976) (stating that when necessary, wiretaps “may be employed to discover the full extent of ... conspiracies”). The appellants’ attempt to bifurcate the “Chicago defendants” from the “Washington-area defendants” thus fails. The government made a particularized showing that the Washington-area defendants were members of a conspiracy whose full extent could be determined only through wiretapping. This showing was sufficient to allow the district court to authorize the government to tap the phones of the Washington-area defendants. The trial court therefore acted correctly in admitting the evidence derived from the taps.
Another of the appellants’ arguments focuses on the response of the trial court to the requests of several of the appellants to discharge their counsel and proceed pro se. Criminal defendants, of course, have a constitutional right to defend themselves. See Faretta v. California, 422 U.S. 806, 818, 95 S.Ct. 2525, 2532, 45 L.Ed.2d 562 (1975). Courts must ensure, however, that a defendant’s decision to represent himself is knowing and intelligent. See id. Thus, the Supreme Court has required trial courts to make defendants “aware of the dangers and disadvantages of self-representation.” Id. In addition, this court has stated that trial courts must engage defendants in a “short discussion on the record” regarding these dangers and disadvantages. United States v. Bailey, 675 F.2d 1292, 1300 (D.C.Cir.), cert. denied, 459 U.S. 853, 103 S.Ct. 119, 74 L.Ed.2d 104 (1982). The appellants contend that the trial court in this case failed to warn them adequately of the hazards of self-representation. The appellants concede that the court informed them in general terms of the dangers of proceeding pro se; the appellants’ argument rests entirely on the failure of the court to discuss the especially complex nature of RICO cases and the difficulties of preparing an unaided defense while incarcerated. We think this argument must fail. The record discloses that the trial court engaged in a full discussion with the appellants regarding the dangers of self-representation. The judge informed the appellants of the seriousness of the charges against them, warned them that he could not assist in their defense, told them that he would have to conduct the trial in accord with the Federal Rules of Evidence and Criminal Procedure, and stated that “it is a distinct handicap to be engaged in a serious criminal matter without any legal training or background and without the active assistance of a trained lawyer.” I J.A. at 1487. In addition, the judge asked the appellants many times whether they understood his comments and whether they had any questions. We regard this colloquy as a model. Neither case law nor common sense supports the position that a trial court must advise a defendant of each and every difficulty he might encounter in a particular case. If the appellants again elect to proceed pro se, the trial court need do no more than it did in the initial trial.
The last of the appellants’ arguments that we take up concerns the Pinkerton doctrine. In Pinkerton v. United States, 328 U.S. 640, 645-48, 66 S.Ct. 1180, 1183-84, 90 L.Ed. 1489 (1946), the Supreme Court held that a conspirator could be found guilty of the substantive offenses of other conspirators if the offenses were committed in furtherance of the conspiracy. The government attempts to support the conviction of appellant Bethea on several counts by reference to the Pinkerton doctrine; the government contends that notwithstanding the absence of evidence that Bethea participated directly in certain offenses, he was properly convicted of the offenses because he was a member of the conspiracy and his co-conspirators committed the offenses in furtherance of the conspiracy. Bethea replies that a conviction may be upheld under Pinkerton only if the court has instructed the jury of that doctrine. We agree. The Supreme Court has stated specifically that the Pinkerton doctrine will not support a conviction unless the court has given the jury a Pinkerton instruction. See Nye & Nissen v. United States, 336 U.S. 613, 618, 69 S.Ct. 766, 769, 93 L.Ed. 919 (1949); see id. at 621, 69 S.Ct. at 770 (Frankfurter, J., dissenting). If the government wishes to rely on the Pinkerton theory on retrial, the government must request and receive the appropriate instruction.
III. Conclusion
We are aware of the length and cost of the trial that took place in the court below. We are aware of the painstaking, conscientious, and sensitive manner in which the district court handled the peculiar problems of the trial. Finally, we are aware of the apparent strength of the government’s case against the nine appellants. But we cannot factor any of these considerations into our decision. The record evidence discloses a real possibility that one juror, for whatever reason, was not persuaded that the government had met its evidentiary burden. The dismissal of this juror violated the appellants’ constitutional right to a unanimous verdict. We must therefore reverse the convictions.
It is so ordered.