Reference / Paper · 1943
CCM (1943–1945) — OP-20-G file, OCR version
CCM correspondence and cycle studies, Dayton–Washington and GCCS links (77 pp., 24 MB) [writer A]
Reference / Paper · 1943
CCM correspondence and cycle studies, Dayton–Washington and GCCS links (77 pp., 24 MB) [writer A]
MATERIAL IN THE C.C.M. FILE The correspondence IN this folder pertains to the feAse of C.C.M.’s (modified ECMs) on the Dayton-Washington link and the GCCS (British) - NCA link. The correspondence pertains prin cipally to: (1) COM Security (2) Procurement and Servicing of CCMs for the above link. SECRET •13 April 1945 Reference: Secret dispatch from Eaohus to Angstrom 10 April 1945. Re CCM Mark II, need total of eighteen each bearing collar and support plate for off-on-encipher-decipher controller switch/ These items wearing badly on four units, interfering with proper operation. Receipt: Received of Naval Code and Signal Laboratory this date: Part Mo. Support plate S. R. OREM, Jr Lt.(jg), USNR 0p-20-G-4-D U. S. NAVAL COMPUTING MACHINE LABORATORY C/O NATIONAL CASH REGISTER COMPANY DAYTON 9, OHIO 23 March 1945. MEMORANDUM TO COMMANDER h.T. EMGSTROM: Sub j: Paper tape for COM - request for. 1. It is requested that one dozen rolls of subject tape be furnished this activity. The present supply is almost exhausted, therefore prompt attention to this matter would be greatly appreciated. R. I. MEADER. U. s. NAVAL COMPUTING MACHINE LABORATORY C/O NATIONAL CASH REGISTER COMPANY ÖAYTON 9, OHIO 23 March 1945. MEMORANDUM TO COMMANDS H.T. ENGSTROM: Subj: Paper tape for COM - request for. 1. It Is requested that one dozen rolls of subject tape be furnished this activity. The present supply is almost exhausted, therefore prompt attention to this matter, would be greatly appreciated. R.I. MEADER. TO: SIGNAL SECURITY AGENCY 1Ü1835Z UAK 45 nsG. NO. GCCS 3293 SSA 535 CIT PHI FOR COLONEL ALLSOPP FRO* St. ALL ULTRA STUDY BY BRITISH ON CC!'. FAULTS HAVE LEEN DIS COVERED IN NOTCH PATTERNS OF ROTO 13 ON ARMY CCBP 0114 (CSF 1814) AND OK NAVAL NETS RRUSi. (CSF 2513 AND CSF 2514) KADY'.. (CS? 1580) AND CSMIZ (ESP 1581). ON CCBP 0114 ROTORS THERE IS POSSIBILITY OF LOBSTE R» DUE TO NOTCHES OCCURRING AS RESULT OF FURTHER (..U) SIMULTANEOUSLY ON EACH SIDE OF ROTORS. THIS OCCURS 10 TILES OK ROTOR NVÖBSR 40, 5 ON 41, 7 ON 42, 10 ON 43, 8 ON 44, 6 ON 45, 8 OK 46, 5 04 47, 5 ON 48, 7 ON 49. ON BRUSA CIRCUIT TO.BE IL FORCE AUGUST 1945 ROTORS NUMBERED 40 THROUGH 49 LOBSTERS OCCUR AS FOLLOWS 7-6-5-37-4-5-7-4-8. ON CUMEC!' BRUSA ROTORS 30 THROUGH 39 AS FOLLOWS 2-5-9-7-1-Z-1O-O-3-4. ON K-DYQ 4-4-4-4-6-9-8-8-5-10 ON CflilZ NUTB31ED 10 THROUGH 19 5-5-5-5-5-7-7-3-7-S. ALSO 338 CYCLE OCCUiiS ON 4 OUT OF 5 WHEELS ..HENEFER CEillZ 10 IS II. POSITION 4 OR 10R IN 2 S3 ICLi. IN CURREUå1 BRUSA WHENEVER ROToR 33R IS IN 4 OK 33 IN 2 SEMICLN ^34 AUGUST 1945 BRUSA WHENEVER ROTOR 43 IS IL’ 3 OR 433 IN 3 OR 47 IN 3 OR 43R IN 3. THESE LAST BRUSA ROTORS ARE IDENTICAL TO FA10US ' UOTE ROTOR FOURTEEN END >UOTE IN THAT ^NUMBER OF NOTCHES AT ODD POSITIONS EQUALS NUMBER OF KETCHES A^feVEN - 1 - POSITIONS. THE FACT THAT CSHTAIK ROTORS IN POSITIONS 2 AND 4 GIVE RISE TO 338 CYCLE OF FOUR WHEELS IS DUE TO SUM OP E3ERY OTHER GROteb OF INTERVALS BETWEEN NOTCHES EQUALLING 13. WAS JUST DISCOVERED BY MAJOR BABBAGE HERE. THIS LAST .WOVE INFORMATION EXCEPT THAT PERTAINING TO CCBF 0114 IS NO DIRECT CONG SRI. OF ARMY BUT AM PASSING TO YOU FOR YOUR INFORMATION AND HAVE TOLD CAPTAIN LESHER. LESHER HERE AT GCCS TODAY. DELIVERED TO BRITISH SERVICES - 2 - EQUIPMENT HE COURIERED ARLINGTON HALL STATION MESSAGE CENTER INCOMING MESSAGE FRC4!: GCCS, LONDON MGLAND TO: SIGNAL .SECURITY AGENCY MSG NR: GCCS 3420 1114122 MAR 45 SSA 544 CRT FRI ULTRA IL GCCS 3293 I GAVE 2 REASONS FOR SHORT CYCLES TO OCCUR IN COM. REASONS «'ME WRONGLY INTERCHANGED. FA. OUS ROTOR 14 ILW ALTERNATE INTERVALS BET17EEN NOTCHES SUITING TO 13. OTHEi ROTORS IN kOSITIOI'S 2 AND 4 HAD NUMBER OF NOTCHES AT EVEN POSITIONS E .UAL TO ODD. BELIEVE BOTH THESE POSSIBILITIES KNOWN TO US LAST SUMMER „HEN WE STUDIED COM. EACHUS RE VESTS COPY WIRE GCCS 3293 AND THIS WIRE BE GIVEN ENGSTROM AT JIP 20 G THES SECRET Op-20-0/mb 29 December 1944. ^rom; 'T’o: 0?-20-0-4. 0? -20-3-GY-A. Subj: oorununication Change - Washington to cayton. X. Commander . I. deader has requested that a dispatch be sent to the 'aval computing ! a chine Laboratory limed lately, before 1 January 1945, ordering the new C.O.M. set-ups into effect. This was requested to circumvent the confusion that occurred during the last change. 2. ill you please comply with this request. H. T. »C • 1 H • TOP SECRET Subject: Possible Machine for Performing Tallying. Enclosure: (A) Rough pencil sketch of subject machine. O.C.M. 1. Sire sufrj'e&fr machine wowfrd operate from four tape readers run synchronously, „ v ________ r two fpr_Jiha_Cdi£ferenced) plain bigram and, two for the cipher,^Zne line-pattern>could h--' run off Unless than two minutes, hence all t eh-in twenty minutes plus time to change tapes. The fifteen patterns involving trigrams would be rather clumsy to handle along these lines, but the difficulty would undoubtedly be overcome under the pressure of necessity. 2. Picture Lt. Levit’s catalogue as a 676 x 240 rectangle, the rows named by the 676 bigrams and the columns by the 20 x 12 combinations of wheel and displacement. A cell of the rectangle is marked if the corresponding entry occurs in the catalogue. In the machine, the rectangle would consist of 676 horizontal and 240 vertical wires connected at the mark ed cells. There are two such rectangles, one for the plain bi gram and one for the cipher. The two impulses from the plain readers (taken to be E and N in Enclosure (A) as illustration) are converted by a 26 x 26 square matrix into a single impulse (for the bigram EN). This enters the plain catalogue board (at row EN) and emerges on the vertical wires connected to this row. The cipher bigram (JV) is treated likewise at the same instant«, 3. The 240 column wires of each of the two boards lead into a 240 x 240 square as shown in Enclosure (A). At each point of intersection (except possibly the ”impossible” ones), there is a counter«of some sort. 50,000 counters of the usual variety would probably not be feasible, but it would be possible to do the job photographically, using lamps costing only a few cents each. S---------------------------------------------------------------------------------------------------------------- GE (originator) 4~.// TOP SECRET T told Lt. Norris that you had agreed with Commander Cross that Ens. Ritchie and Lt. Anderson night be available part time on this work. As far as I know, nothing farther has happened on this. FRIEDEL CALCULATORS FOR LT. COMDR. NENZEL. A request from Lt. Comdr. Nenzel appeared for two Frieden calculators. This memo was addressed to GO via Ct-50 and 0-20. mhe memo had previously been approved by 0-20. I thought you would not like such tactics and called GE to ^et their opinion of Nr. Nenzel’s needs. They said that Ir. Nenzel kept his present calculating equipment very busy and felt that he was warranted in asking for two more— particularly since GE might then be able to use them part time. Since G-20 had already approved the purchase, I ap proved the memo for you. CALL FROM LT. I. R. TRITE. ‘Bob White called from Dayton to say that he was only working about two hours per day and was anxious to be re assigned, but that the matter could most certainly wait until you returned from leave. BILL TRAY CLASSIFIED 1A. Bill "ray said that he had just received notification that he was reclassified 1A. I took the matter up with Commander Foley who talked with someone in GR about the past history of this case. Commander Foley said that Bill Uray, must have been classified as 1A for social pur poses by friends on the local draft board. He says there is a rule by which it is not easy to ask for deferment for civil service personnel on the grounds of indispensable work. T7e said that they could probably break this rule but that the memo might not be approved and might bring undesir able attention to Bill "ray's case. TTe wonders if the best bet is simply to let the matter ride on the basis that "ray will probably never be called for induction, "e asked me to sound around to see what Bill "ray's feelings were in the matter and whether it was worth saving his pride, believe that "ray feels rather or the defensive inasmuch as if his work here is rot worth asking for a deferrment, he is not doing the job he should do. In view of your experience on this matter in the past, I thought I had just better let it ride until you return. 1 COMMITTEE MEETING The .Joint Army-Navy Committee Meeting was postponed until ' 'ednesday, 27 September. COM COMWMCE. Lt. Comdr. Linn and Lt. Levit were aboard Monday, IS September for conference with Lt. Clifford regarding the COM. The conference was held in Commander Ford’s conference room. DATTOM COMTPLETEMT. Commander Foley, Lt. Comdr. Latta, and various people in GR have been heckling us for exact information in regard to what is going on in the way of the assignment of the ’"ave complement in Dayton. Andy felt that the matter should be straightened out with a memo to all concerned and has a tentative draft prepared along this line for your approval. ENSTGM NITES ABOARD. A new engineering officer, Ensign Miles, reported aboard and was assigned to GJ. He has had no particular radio ex perience, but has worked on electronics and has evidently held some responsible Jobs, including design of equipment for a power company and electronic demonstration equipment for his Alma Mater, Motre Dame. STATISTIC AI. BOMBE. Lt. Clifford’s letter to Eachus last week included a rather complete write-up as to what we were trying to do in the way of a statistical bombe. I asked Commander Roeder about this in regard to whether it should be revealed in a more or less official channel at this time, and Commander Boeder said simply, "why not omit it this time and wait until Commander Engstrom returns". I told this to Lt. Clif ford and he was very annoyed because Commander Reader presumed to censor his letters to Joe. TJe said that if it kept up, he would write Joe directly. I believe that-^calmed him down on this and told him that it was my fault that the item was omitted this time and explained that the policy top-side was very indefinite in regard to revealing new matters and therefore we had to be rather careful about it. He saw the point of this but T perhaps was a bit over-cautious in regard to this particular item. PHOTO RADIO MOVE. Lt. Morris received a chit from Commander Cross which stated that Commander Arps, evidently without consulting us, had instructed the Bureau of Ships to move the photo-radio set-up from Skaggs Island to San Francisco. Lt. Morris evidently feels the natter should wait until Commander "enger returns. STATISTICAL BOMBE. We are still pushing ahead on means for obtaining a statistical bombe. Lt. s,1g) Orem is drafting a letter to the GE representative here for someone’s signature here which would declare our interest in Selsyn motors and would state what priority we intended to use to back such an item. Evidently the Army Signal Corps has about 9,000 sets bein" manufactured which would suit our needs to a tee. Perhaps we car hook onto one of these sets. We are also working strongly on a dudbuster application on Granddad. T e are now in the process of weighting .lackpot and random tries and "Pete Geffert will sum and tabulate a series of these tries for us on I.B.N. equipment. MOVE OF WARMER’S EOTTIPMTNT, Equipment on the stage and in the room below are being moved today, Thursday, 21 September 194-4. There was con siderable feeling that Tetra "Vessie" should be moved at this time, not only to take advantage of the trucks avail able, but also to make room for a dark room for Wilson. I vetoed this suggestion, said I felt that moving of the photographic equipment should wait until we have more stable plans along this line. It may be quite difficult to move metra "Tessie" because at the time it was moved in there were no permanent bulkheads in the passageway in the old gymnasium leading into the school building. YWTG ENGINEERING OFF IC ERS. Lt. Norris says that some of the officers that have been assigned to him have rot exhibited any great degree of imagination in their work. He feels that they might be better fitted for maintenance. Perhaps they are .lust wash outs. Blakely has quite a few officers on ice; perhaps now is the time to consider a shift around of these officers to more permanent duty. As I remember, Meader was supposed to get some of these officers; perhaps his needs should be con sidered also at this time. Leader feels that Calmer is way and above most of the other officers from a research stand point. nerhaps Meader might be agreeable to have palmer come’ here and take charge of the research and development and also the laboratory if Rowley goes on field work. T' is appeals to me very much as a solution to a number of our problems, particularly if the Navy is perhaps being squeezed out of the research work at the National Cash Register go. I believe that Palmer would be able to hold his own on this job with respect to both Steinhardt and Blakely’s group. HABELIN ,tqb ppp ARMY. Captain Seaman brought over a 2700 character tape and asked if we could run six (6) strippins; jobs on,MIKE for the Army. r’e did this for him on one midwatch and the results (4* have been sent over. He said that the man in charge of this job was very interested in MIKE and that this was a test run to see how effective MIKE was on this type of work in compari son with I.B.M. methods which I understand takes several days to do. VIPER BANKS FOR ARMY ■ You will perhaps remember that you told Colonel Rowlett you would find" out what we had available in the way of thirty (30) VIPER stepping switches that he might use in the construe tion of two (2) Furpie machines. Perhaps you-might want to take this up with Meader while he is here. MAMBA. Lt. Comdr. Meader reports that he became very annoyed with the slow progress on MAMBA at the Acme Pattern and mOol Company. He went over there a wetjk ago and told them that they were laying down on the job**^ince they had all detail drawings dore, He wanted every part in the process of manu facture simultaneously in their place. He assigned Bateman for a half-day duty each day at Acme to expedite MAMBA. He feels that MAMBA will still not be available for from six to eight weeks. This is too bad in view of the feeling in CAr-‘r that MAMBA should have been done long ago and their apparent strong desire to break into the type of IM-2S traffic that uses key additives in the indicator system. CHINESE DIPLOMATIC TRAFFIC. Commander Holdwick called, me in to see who was working on Chinese diplomatic traffic. I found that you had given the job to Bill Wray and that Bill had only the original set of messages and under your instructions had more or less secured interest until further traffic occurred. Bill "ray and I went to see Commander Holdwick and Command er.Wright and they were evidently interested in doing something about this. They are of the impression that a third large folder is available somewhere and I believe, they are trying i>o track it down. Bill Wray can tell you more of the details in regard to this matter. SECRET ZWU ASH REGISTER OFFICIALS ABOARD. Fr. Allen and another official of the National Cash Register Company held a conference with Captain Kinney and Lt. Comdr. Leader on Thursday, 21 September 19LA. The con ference is going on as T write this so I don’t know what is happening. I believe the subject under discussion is in re gard to sending spare parts of Bombes here. FAT BREAK. The British got into the NAT traffic this month by the "X” method. The break they got was on a wheel motion that astounded Alexander in view of their work on the traffic on another day. We do not know yet whether Alexanders figures were wrong or that the motion changes frequently. Alexander and Gleason will continue with the post-mortem trial on it. Icky ran their first problem with a fair degree of success, but without finding the jackpot -- possibly because the wrong motion was assumed, ü) fytvnA**' SEMI-AUTOMATIC PROGRAM. ^he Captain saw the Admiral on Wednesday and then called Lt. Porris in to say that the Admiral wanted a complete review of this situation together with comparison between teletype and letterwriter semi-automatics. At Norris’ request, I put Lt. Rowley on this job with Lt.(jg) Patterson to help, ^oday Commander Wright called and requested the original program for semi-automatics. I took him a copy of the memo I wrote you settir." up the next year's order on T.B.TL LADE RIP. Lt. Comdr. Raven finished his RIP on the -TADE and I signed for it. It is in your file. WRITE-UP ON ENGSTROM FAT’S. Ten (10) copies of a write-up on Engstrom Pans are in your file. CONFERENCE OF WAVE PROPAGATION. ^he following was received from the Wave Propagation Committee or which you might wish to take action: Plans are being made for third conference to be held at 9:30 A.M., 16 and I7 November 1944, in Auditorium, National Academy of Sciences, 2101 Constitution Ave., ’’’ashinmtor, D. C. mhis notification asks that information be supplied as to who would attend and who might submit reports, etc. T 0 P S E C R 75P-2O-GE7m5 d MM"s 24 July 1944 MEMORANDUM From: To: OP-20-GE di OP-2O-G5O f‘ Subject: C.C.M., termination of researches on at Arlington Hall. 1. The technical Sub-committee of the Committee to in vestigate the security of the C.C.M. met for the last time at Arlington Hall on 22 July. The first meeting was on 1 May. The projects completed during this period are outlined below. Lt. Levit (0P-20-K) is continuing with a project of his own, and I shall maintain contact with him and Lt. Cmdr. Linn. The Army contingent is, however, securing. A report will be made shortly to the Committee. 2. The Sub-committee will make the following suggestions to the Committee: (a) Rigid enforcement of'the ”legal limit” of 200 groups to a message. (b) Change the indicator system to one not exhibit ing monoalphabetic characteristics. Thus, under the present system, two messages in the same day with indicators LJCAQ and XTCRQ have the same initial setting of the Z- and 5-wheels. (c) When the 3-wheel has a stepping contour on both sides, all wheels will turn ("lobster") one time in four. This may be remedied by avoiding oppositely placed contours in making up the wheels. Actually, OP-20-N has already made a set of "anti-lobster" wheels. (d) It is unwise to simply rewire a compromised set of wheels, without changing the cam contours. In destroying wheels to rrevent capture the cam contours must be obliterated as well as the wiring. 3. The projects leading to the above suggestions follow. Except in (c), it is assumed the enemy is in possession of a C.C.M. and the appropriate set of wheels. (a) The wheel-order and setting of a 1000-group mes sage (actually sent), with matched plain and cipher text, were found by means of lobsters. The same was accomplished with a W secret day’s file of 250 messages all within the legal limit of 200 groups; to do this the indicators were used to line up mes sages to be in depth on the 3-wheel. (b) From the one set message in (a), the daily in dicator setting was found by an IBM sort. Thus, with known wheel order, a five-letter crib is easily set. (c) Using the long message in (a), and assuming the stepping pattern, which can be found from the lobsters, the wiring was recovered. (d) Two 200-group messages were enciphered with all but the 5-wheel in depth, and having a 7-letter plain text co incidence between them. Using the cipher text only, the wheel order and settings were found, and the messages read. The technique used was statistical, and a machine similar to Hypo could handle it. Without the 7-letter coincidence, however, the number of trials would be about 2| billion, and there is some doubt whether the right answer would stand out. 4. Lt. Levit’s project is to find the wheel-order and setting of a single message of 300 groups, using the period of 338 on the center three wheels and the constant precession of the end wheels. (These were also used in project 3 (c) above). His method becomes unworkable at the legal limit of 200 groups, and consequently no recommendation is contemplated at the present time to avoid the center cycle of 336. 5. Needless to say, consideration of ”bad" wheels, which produce a short cycle, was excluded. 6. I believe that the results of paragraph 3 show that, unless the suggestions of paragraph 2 are adopted, there is a small but non-negligible chance that the enemy (having captured a 0.0.M. and wheels) might read an occasional day. The use of the 0.0.M. for Ultra dispatches of the bean-spilling variety would then plainly be unwise. On the other hand, it is note worthy that none of us thought of an operationally feasible method, whereby the enemy could read more than a scattered day here and there. The difficulties are: (a) (b) (c) ' I close to a million wheel-orders; erratic stepping of the end wheels, the one regularly stepping wheel being well buried; no discoverable test on whether a stretch of plain and cipher text is properly aligned (such as by crashing on enigma). Because of (a), bombing is out of the question, and because of (b) it is hard to see how to strip off wheels one by one. ' feel that the designers of the C.C.M. did a very good piece of work. TOP Respectfully submitted, A. H. Clifford Lt. U.S.N.R. 0P-20-G/rh TOP SECRET 17 July 1944 MEMORANDUM Fron: To: Via: 0P-20-GY-A OP-20-GM ,A^ 0P-20-G-50 (fl Subj: COM, Installation of Page Printer in Circuit of. 1« Confirming telephone conversation of this morning, it is requested that there be inserted into the circuit of the COM used in this Section for traffic with the British a page printer to permit outgoing and incoming traffic to be placed in final form immediately upon decryption. R. B. ELY LT. CDR. , TTSNR L4z 24 June 1943 Date: From: To; , Prof.. Bayly .■» W. “ - -- ... - Captain Stone Subject; Modify Ing request of 5 Ibn. Request equipment for_use_w_.t __ CCi - yacjiine Hied Under: «IWK SECRET 0?-20-GE/mb 17 May 1944 MIWRANDUM From To : : OP-20-GE .L O?-2O-G5O^> Subject : Cross-tie In C.C.M. from 1-Wheel To 4-'Tieel. Enclosure: (A) Secret Monograph entitled. "Effect of Cross tie on Period of C.C.M." 1. On 4 May 1944 a modified C.C.M. basket, vå th a cross-tie from the 1-wheel to the 4-wheel, was delivered to this office. This device causes the 4-wheel to step when an indent comes up either on the left side of the 1-wheel or on the right side of the )-wheel. Enclosure (A) represents our analysis of the effect of this device upon the period of the 4-wheel, and hence upon that of the middle three wheels and the machine as a whole. 2. In the unmodified machine, the period of the 4wheel is 333« With the cross-tie, and using normal wheels, tais period is never 338, but is either 4394 or 57122, depend ing upon the choice of the 2- and 3-wheels, and is the larger of these for at least 90% of the wheel choices. The period of 338 of the middle three wheels is tnus completely destroyed, and the period of 4394 of the wnole machine is stepped up to 57122 at least 90% of the time, not considering the benavior of the 5-wheel. We feel, therefore, tnat the cross-tie admir ably disposes of the weakness of the C.C.M. due to the period of 338 on the middle three wheels, when normal wheels are used. 3. If defective wheels are allowed, the 4-wheel may, for a given wheel order, nave periods of 338, 676, 4394, or 8788, depending on the initial setting. Although the two larger values hold on the average for at least 90% of the settings, the fact remains that there is a chance of up to 10% that a long message will hit a short cycle. In this event the 5-wheel will at best precess a fixed amount every 338 steps, and this situation may be as bad as true depth. It is our opinion, therefore, that the use of the cross-tie with defective wheels will greatly reduce the risk, but that since it is not reduced to zero such use cannot be classed as "secure". Respectfully submitted, Alfred H. Clifford Lt., U.S.N.R. SECRET (Si-E/mb ■ MM EFFECT OF CROSS-TIE ON PERIOD OF C.C.M. 1. /{MAY MH Summary of Results The device under consideration is a tie from the left side of the 1-wheel to the 4-wheel, so that the latter will step if there is an indent either on the right side of the Z-wheel (as usual) or on the left side of the 1-wheel. Its effect is considered with relation to the following three facts concern ing the C.C.M. when all wheels have an even number of indents. (1) The period of the middle three wheels is 338. (2) The period of the whole machine is normally 4394. (3) When ”defective” wheels are used, the period of the whole machine may be only 338. Thus the setting AAAAA. will become say KAAAS after 338 steps, the two end wheels precessing a constant amount, always even (here 10 and 18 respectively), and thus returning to their initial positions after 13 x 338 - 4394 steps. The cross-tie will evidently have no effect upon the 1-, 2-, and 3-wheels. We consider only its effect upon the 4wheel. The behavior of the 5-wheel would require much deeper analysis. If the cross-tie is used with normal wheels, it is found that the 4-wheel can never have a period of 338. It does, however, have what we might call a ”statistical period” in the sense that there is a certain probability, ranging up to about one chance in fifteen, that it will have the same setting in any two particular machine positions 338 steps apart. Sm.ce this probability is so low, we are justified in saying that the cross-tie effectively breaks up the period of the middle three wheels. The same is true if the right side of the 3wheel is defective, and consequently any short cycle on the 3-, 4- , and 5-wheels is destroyed. The effect is somewhat different if the short cycle occurs on the 1-, 2-, and 3-wheels. In this case, for a given wheel order, certain initial settings may produce a short cycle even with the cross-tie in operation. The 263 settings of tnese three wheels fall into 52 distinct cycles of length 338. One expects three or four of these to be "bad”, producing a cyc^e of 338 on the 4-wneel, and nence on all except possibly the 5- wheel. In a ”good" cycle, the 4-wheel can never return to its initial setting after 336 steps. -2- Using the cross-tie with normal wheels, the 4-wheel pre cesses a certain even amount d after 4394 steps. This amount is zero if the left side of the 1-wheel has the even-odd defect, in which case there exists a period of 4394 on all except possibly the 5-wheel. If, however, the left side of the 1-wheel is normal, the question of whether or not d = o depends wholly on the 2- and 3-wheels. It would be practi cable to find it out experimentally for the 20 x 18 = 360 possible choices for these two wheels, in a set of ten wheels, but no simple criterion is apparent. The proportion of ”bad” choices appears to be the random 1 in 13. When d - o, the period of the first four wheels is 13 x 4394 - 57122. When defective wheels are used, and a short cycle exists on the 1-, 2-, and 3-wheels, then in a "good" cycle the 4wheel precesses a certain amount d, not zero and possibly odd, every 338 steps. Consequently the first four wheels will have a cycle of 4394 or 8788 or 676, depending on whether d is even, odd and not 13, or 13. The number of these various cases, in cluding the number of ”bad” cycles for which d = o, is easily found for a given wheel order. With say, four defective wheels in a set of ten, they could be tabulated for the 4 x 12 x 10 - 480 possible choices of the 1-, 2-, and 3-wheels. In the examples, fictitious indent patterns are used. For convenience, we assume throughout that when a wheel steps forward, the letters indicating its setting increase alphab etically. 2. Effect of Cross-ties at .intervals of 338. Without the cross-tie, the 4-wheel returns to its initial setting after 338 steps. With the cross-tie, the 4-wheel will receive a certain number of ”extra boosts” from the 1-wheel when the right side of the 3-wheel is in an inactive position. To find the displacement of the 4-wheel after 338 steps, we need only count the number of these ”extra boosts” that it has received. In Figure 1, assumming the indent patterns for the left side of the 2- end 3-wheels as shown, 338 successive settings of the 1-, 2-, and 3-wheels are given, starting at AAA. Those of the 3-wheel are the normal alphabet, repeated for each column, and are omitted. The 339th position would be YAA, and the next 338 settings differ from those shown in Figure 1 only in that the 1-wheel is two steps behind (Y instead of A, Z instead of B, A instead of 0, etc., throughout). In Figure 2, indent patterns are assumed for the left side of the 1-wheel and the right side of the 3-wheel. The 3-wheel is in the inactive setting E just 13 times during this block of 338 steps, and reference to Figure 1 shows that the 1-wheel -3- is at these times in the settings BOZLCKUC£ITDO Of these, the settings D, L, O, Q, and T (underlined) are active positions of the 1-wheel, and the rest are inactive. Consequently the 4-wheel receives six extra boosts while the 3wheel is in setting 2. By adding up the boosts for all the inactive positions (E, F, I, J, N, Q, S, T, L, and Z) of the 3-wheel, we find the desired displacement of the 4-wheel after the 336 steps shown in Figure 1. For this purpose a frequency count is made of the 10 x 13 = 130 letters (settings of the 1wheel) in these rows of Figure 1. The result given in Figure 2. One extracts from this count the active settings of the 1wheel: A D G H L 0 P Q T 4 4 4 2 4 10 3 3 7 3 44 Since the total is 44, the 4-wheel has received 44 extra boosts in these 338 steps. If it starts at A it will end at A + 44 = A t 13 = S. If we started at position ZAA-- instead of AAA— in Figure 1, every letter giving a setting of ehe 1-wheel is diminished by one alphabetically. If we made a new frequency count for the rows in which the 3-wheel is inactive, we would find as many A’s as formerly there were B’s, as many O’s as formerly there were E’s, etc. Hence we can find the number of extra boosts as the sum of the original frequencies of B, 0, H, I, etc. These are entered in Figure 2 in the column headed Z, and the total is 41» Since, in the example, the 1-wheel moves from A to Y in 338 steps, then to W, etc., the position of the 4-wheel is found from the upper line of totals (44, 56, 44, 49, etc.) in Figure 2. The successive positions of the 4-wheel after each batch of 338 are tnerefore as follows: A«-44 = A*18=S + 56 = S * 4 V/ ♦ 44 = W - 8 0 + 49 = 0 - 3 1* 55 = 14- 3 = 0 3 0 + 39 -0-15-8 B * 55 = B + 3 = E E ♦ 52 = E + 0 = E E «■ 51 = E - 1 = D D ♦ 40 = D * 14 = R R * 54 = R * 2 = T T * $2 = T * 0 = T T * 49 -T - 3 » Q Twice in the above 13 times the 4- wheel returned to its starting point (E and T respectively) but there is no evidence of a period. Figure 3 indicates the absence of a period a lit tle better. Here the settings of all but the 5-wheel are given for the first 31 positions of successive batches of 338 steps. Active positions of the left side of the 1-wheel and the right side of the 3-wheel are indicated by arrows. Of the 62 com parisons made here, between a given position and that 338 steps later, just 4 have the middle three wheels in the same position (IÜIC, IVL, I'M, and LXN). These are all in a lump, comprising ”Zone 8”, A zone is a stretch of active positions of the 3wheel. Within a zone, the displacement of the 4-wheel is the same. If one begins at CDF— instead of AAA— in Figure 1, the frequency count is altered only in that one Z is added and one B is removed. Beginning anywhere in Zone 2 (CDG-, CDH, or CEI), one adds an A and removes a C, in addition to the above Z - B exchange. Proceeding thus from zone to zone, we get the frequency changes noted in Figure 2. Increases or decreases in the resulting totals are shown by + ’s and -’s. Places where the total equals 52 are encircled. Considering the width of the zones, there are 44 such repetitions in 676 comparisons, as seen in the following tabulation: WIDTH 5 1 3 1 4 3 2 1 4 2 ZONE 0 1 2 3 4 5 6 7 8 9 NO. 52’s 3 1 3 0 0 0 0 1 3 3 TOTAL 52’s 15 1 9 0 0 0 0 1 12 6 44 From the manner in.which the totals change it is clear that a true period of 338 could exist for the 4-wheel only in the absurdly trivial case in which the 1-wheel had an indent in every other position. The example chosen indicates 44 chances out of 676, or 1 chance in 15.4, of the 4-wheel being in the same setting at two positions 338 steps apart. Considering the selection of 10 letters of the frequency count as a random matter, the probability that the total thereof is 52 is '3<2 yz I ^6/ ' } /y, y MWZO or very close to 1 in 15. Here the mean sum is 5 x 10 = 50. With 12 inactive places on the 3-wheel, it would, be 5 x 12 = 60, and. the probability of hitting 52 (or 78) considerably less. 3. Effect of Cross-tie at Intervals of 4394. To see what happens to the 4-wheel after the normal cycle of 4394 steps when the cross-tie is introduced, we observe that it will be advanced by an amount equal to the sum of every other column of Figure 2 (44 > 56 f 44 + 49 + .. or 41 67 & 44 * 44 .. ). In general terms, let 4 be the number of notches on the right side of the 3-wheel, so that 26 - k is the number of inactive places. Then the total of the frequency count (130 in Figure 2) is 13 (26 - k) = N, and N is divisible by 26 since k is even for the type of wheels we are considering. Let Ne and No be the totals of the even and odd terms; in the example Ne=7 *4+2+ ... = 60 No = 4 * 11* 3 * ... =70 Let a be the number of indents on the left side of the 1- wheel in the even positions, and b the number in the odd positions. In the example a = 6 b = 4 (D, H, L, P T, X) (A, G, 0, Q) Then the desired total is aNe * bN0 s (a-b)Ne * bN = (a-b)Ne (mod.26) In the example, (a-b)Ne** 2 x 60 - 120 s 16 (mod.26). Thus A will change tö A r 16 - Q after 4394 steps, as already noted above. If one started at BAAA- instead, one would get the negative (mod.26) of this. If the left side of the 1-wheel has the odd-even defect, namely a = b, then the 4-wheel will always return to its initial position after 4394 steps. In the contrary case, it depends on whether or not Ne is divisible by 26, or actually by 13, since a - b is even. This number Ne depends only on the indent patterns of the wheels, and not on the initial setting. For the 1-, 2-, and 3-wheels still have the period of 4394. Or, viewed otherwise, the successive changes in the frequency count consist always of adding one even letter and taking another away, or adding one odd letter and removing another odd letter. N could be found experimentally for each of the 360 = 20 x 18 choices of 2- and 3-whsels, in a set of 10 wheels, but no a priori way of avoiding its being divisible by 13 is apparent. If Ne is not divisible by 13, then the 3-wheel evidently precesses the amount (a-b)Ne every 4394 steps, coming back to its original setting only after 13 x 4394 steps. 4. Effect of Cross-tie When Short Cycle Exists. In Figure 4, one indent on the left face of the 2-wheel has been changed (Z to W) in order to produce a short cycle. In Figure 3, the appropriate frequency count and totals are shown. In this case, since the 1-wheel returns to its initial setting after 3)6 steps, there are no changes to be made in the frequency count as we progress along the same cycle. In other words, we find that the 4-wheel gets 47 boosts in the 336 steps starting with AAA—, so that AAAA- will become AAAV-, and we see that uhe 4-wneel also gets 47 boosts if we start at CDF—, or anywhere else in the same cycle. The 263 initial settings of the 1-, 2-, and 3-wheels break up into 32 cycles of length 336. For each of these cycles, the 4-wheel precesses a constant amount d every 338 steps, and d may have any value at all from 0 to 25. For the cycle starting AAA, d = 47 = - 5. In Figure 5 the totals give the values of d for each of the 26 cycles starting at AAA, BAA, ..., ZAA. The values of d for the other 26 cycles starting at ABA, BBA, ..., ZBA can be found similarly. The expected proportion of ”bad” cycles, with d = o, seems no different than the theoretical 1 in 15 occurring above, but will of course fluctuate widely from wheel order to wheel order. In the example, not a single bad cycle exists, at least for half of the 52 cycles. The period of the 4-wheel will tnu’s be 338, 676, 4394, or 8783, depending on whether d is o, 13, even, or prime to 26. Note the occurrence of d = 65 13 for the cycle starting GAA in the example. 338 SUCCESS iW POSITIONS OF 1- AND 2-WHEELS. FIG. 1 21 A B C D E F G H J K L M N 0 P Q R s T u V W X Y Z X X 3£ X X X X X X X X X X X X X X 12 12 12 12 12 12 12 12 12 12 12 12 12 AA AA AA AB BC CD CD CD CE CE CF DF EF FG FH GH HH KK EK MK NL OM PN PN PN PO QO RP RP RP RQ RR RR RR RS RS RS RS RS RT ST TU UU VU WU XU YV ZW ZX ZX ZX ZY ZY ZZ AZ BZ CA CB DB EB FC GC HC 1C JC KD KD KE KE KE KE KE KF LG LH MH NH DI 01 OJ OJ OJ OK PL QL RL SM TM UMVM WM XN XN XO YO ZO AO BO CP CQ CR CR CR CS CS CT DT ET FU GV HV IV JW JW JW JW JW JX JX JY JY JY JY JY JZ KA KB LB MB NC OC PD PD PD PE PF QF RF SG SG SG SG SG SH TH Ul Ul Ul Ul Ul UJ UK VL WL XL YM ZM AN AN AN AO BP BP BP BQ BQ BQ BQ BQ BR BR BS BS BS BS BS BT CU DV EV FV GW GW GX GX GX GY GZ HZ IZ JA JA JA JA JA JB KB LC MC NC OC PC QD QE QF RF SF TG TG TH UH VH Wl WJ WJ WJ WK XK YK ZK AK BL CL DM EM FM GM HM IN IO JP JP JP JQ JQ JR JR JR JS JT KT LT MU NU OU PU QU RV SV TW TW TW TW TW TX TY TZ UZ VZ WA WA WB XB YB ZC AD AD AD AE AE AE AE AE AF BF CG CG CG CG CG CH DI DJ DJ DJ DK EK FL GL HL IM JN JN JN JO KO LO MO NO OP OP OQ OQ OQ OQ OQ OR OS OT PT QT RU SU TV UV VV WW wx wx wx WY WY WY WY WY WZ xz YA YA 1 1 X X X 1 1 X X X 1J 1J IK JK ■FIG, 2 IL FREQUENCY & CHANGES ZR X X X X X X X X X ABD: SUB: ZAAADFGGGH B C C C F H . I I I J 5 11 4 2 4 4 8 5 5 2 1 2 ............... 2 ......4 5 6 7 7 ...2 1... 1 2 ......7 6544 12 11 ..........6 ................... 4 2 X X X 8 T U V W X Y Z X 4 8 X 8 X CONTINUED- FIG. 2 continuedP J I H 7 12 1 4 10 3 3 3 7 4 8 3 4 3 6 4 7 11 2 4 55 51 44 6 2 3 7 1 4 6 4 4 2 39 2 10 1 7 3 3 4 10 3 4 5 7 7 3 8 1 4 3 7 7 5 8 1 4 7 4 3 5 6 8 7 11 4 4 3 2 6 4 7 11 4 3 2 4 4 4 7 11 3 2 2 4 4 7 11 3 6 2 2 7 12 4 3 7 12 6 4 , 1 2 10 >(52) 51 55 (52; 61 36 53 5 1 4 6 4 4 2 7 4 3 7 8 6 4 3 2 7 12 4 3 — 4 4 4 4 4 4 4 V U T 11 4 2 4 7 12 6 12 2 10 7 3 5 7 7 5 8 1 6 4 56 67 41 3 2 6 4 3 5 7 4 3 2 7 4 1 3 7 4 1 4 11 4 12 1 2 7 4 1 8 6 2 6 2 10 5 1 8 3 4 3 4 4 4 2 4 10 3 3 7 7 3 2 7 1 3 3 7 4 4 — 4 + + + — + 4 4 4 Q W Z 44 R X A Y — 4 51 37 4 + 4 + 4 4 49 44 4 — — — — 7 44 — — — — — 4 4 4 4 4 — 50 43 53 73 36 — O 4 6> 4 + — — — 4 4 4 S •• — 4 — 4 4 G 4 51 47 51 0 — 4 4 4 — — 4 — 4 — 44 53 61 L N — — — — 4 — 4 — 4 4 G 52 — 4 0 K 4 4 4 4 <— — 0 — — 4 4 — 4 0 51 52 4 ® 4 — — 4 4 4 4 4 4 - — 4 — — — — — — 4 42 53 F E D c B 6 1 8 4 4 3 6 11 4 7 4 3 2 4 7 11 4 3 11 2 4 7 4 3 2 6 2 7 12 4 1 6 2 12 7 4 2 10 1 6 12 4 10 1 2 6 4 3 2 10 3 7 5 3 8 zone 1 7 4 3 (52) 49 59 j 0 54 64 35 53 4 4 39 — — 40 G 0 — — 4 4 4 4 4 •• 4 4 4 — 62 — 4 47 ) 56) 4 36 4 1 2 — 3 — 4 — 5 6 4 4 7 8 4 9 4 j 10 — — 56 52 53 — 4 FIG. 3 NiP98 N4676 12 9 4 5 1 2 9'4 5 Vi W W W W/P x\ K YvQ ZV A Av8 A Bvl A C?U B dJv Z C D F Vi D sXX D ■A E -A E J A F K\B L\C B C F O G N^E H H F H Q*H G 8 I G T J /G IKK <G /G J WvM X*N G K Y\° H K Z^P D D A\R K BvS K OCT N 0 M E V D K BCR K K 0v3 DvT M E^U N ZONE 2 2 2 4 4 4 5 5 5 A