SOLVING CIPHER SECRETS

N the annals of cryptography, as in any historical record, some dates stand out more prominently than others in the relative importance or interest of their happenings.

Thus A.D. 1586 marks a high spot in cipher history. For in this year there was published at Paris the first edition of an epoch-making treatise on cryptography, " Traicté des chiffres, ou secrètes manières d'escrire," by Blaise de Vigenère, French physicist.

Vigenère was born in 1523, and lived until 1596, ten years after the appearance of the book just mentioned. He was a scholar of note, and the translator of many scientific works into French from the Greek and Latin.

But had he done nothing but write his book on ciphers, in this alone would he have set an enduring monument to posterity. Indeed, the invention of a single cipher described in this work was sufficient to have made him famous.

The cipher referred to is his celebrated chiffre carré—square cipher—a system of far reaching importance, destined for use throughout the civilized world, and long believed absolutely undecipherable without the key.

The cipher consists in the use of a series of cipher alphabets, the identity, number. and order of the alphabets being determined by a literal key, usually in the form of a word.

This cipher of Vigenère's is modeled after, and is a simplification of, a still earlier cipher first published in 1563, the invention of a Neapolitan physician, Battista della Porta.

The Porta cipher is accredited the distinction of being the first cipher ever devised using a variable literal key. Like the Vigenère cipher, it, too, was used by many world celebrities, but somehow it never reached the peak of popularity attained by its derivative system.

Perhaps this was because the Vigenère table. If not at hand, could be more easily reconstructed from memory. Again, the Vigenère cipher may have appeared to offer greater security. For it permits any letter to be represented in cipher by any letter of the alphabet, while the Porta cipher only provides half this number of substitutes.

The Vigenère chiffre carré—also chifire quarré—has more names than the proverbial cat has lives. The French called it chiffre indéchiffrable—undecipherable cipher—and the Germans named it multlplikations-chiffre.

It was used as an official and military cipher by the Confederate States of America during the Civil War, hence the name Confederacy cipher.

The Confederate key had the additional feature—not shown in the accompanying table—of a column of numbers, 1 to 26, from A to Z at the left side of the table; and a row of numbers in descending order, 26 to 1, from A to Z across the top. In this way, presumably, the alphabetic square could also be used with a numerical key.

Incidentally, it is said that cipher keys exactly similar to that used by the Confederates were in the possession of John Wilkes Booth and his co-plotters, one copy having been in Booth's waistcoat pocket after he was shot.

It was supposed that these keys were supplied by the Confederate government. But it cannot be assumed from this that the Confederates had any previous knowledge of the plot to assassinate Lincoln.

The Vigenère cipher was profound in its influence upon subsequent systems. The Porta multiple alphabet principle, through it, became tremendously popular. And the chiffre carré became the model for numerous ciphers, many of which were identical with it in results, but different in manipulation.

One of these is the Gronsfeld cipher—see FLYNN'S for June 6, 1925—and another is the St. Cyr cipher, said to have been used by both armies in the Franco-Prussian War.

The chiffre carré is also variously known as the Russian square, alphabetic square, multiple-alphabet and block-alphabet cipher. Continental cipher, pass-word cipher, and Sphinx cipher.

Vigenère's cipher, as already mentioned, is poly alphabetical, the whole number of cipher alphabets being equal to the number of letters in the alphabet of that language for which the alphabetic-square is constructed.

The table for the English alphabet is given herewith. Each of the twenty-six horizontal cipher alphabets takes the name of its initial letter, also found in the key alphabet at the left of the table.

Thus the cipher alphabet in the first row, beginning with A, is the A-alphabet; the next one below it, the B-alphabet; and so on.

The substitute in any alphabet tor a given letter is found at the intersection of the desired alphabet with the column of the desired letter. Thus the substitute for K in the B-alphabet is L; since this letter is found at the intersection of the B-row and K-column.

In some of the Vigenère alphabets may be recognized previously known single-alphabet ciphers. Thus the Augustus cipher—in FLYNN's for February 21, 1925—is identical with the Vigenère B-alphabet. Julius Caesar's cipher—in the September 12 issue —is the equivalent of the D-alphabet here. And the Albam cipher—see the issue of October 10—is the same as the present N-alphabet.

From this it will be seen that Vigenère's cipher does not differ in the formation of its individual alphabets from those of simple ciphers already centuries old. Its difference lies in the fact that it uses a series of such alphabets, the identity, order, and number of which is only known to those having the key.

To illustrate the method of using the Vigenère cipher, the short message at (b) will now be enciphered with BASTILLE as a key:

(a) Key:     bast illeb ast ill ebasti lleb as tilieb asti llebas
(b) Message: KING LOUIS AND HIS FAMILY PLOT TO ESCAPE FROM FRANCE.
(c) Cipher:  LIFZ TZFMT AFW PTD JBMAEG AWSU TG XANLTF FJHU QCEOCW. 

First, write a letter of the key above each letter of the message, as at (a), repeating the key as the length of the message requires. Each letter of the message is now enciphered in the alphabet indicated by the key-letter with which it is paired, in the manner already described, the completely enciphered message being shown at (c).

In deciphering with the key, the process is reversed. Thus to decipher the first letter L of the above cipher, locate the B-alphabet, follow it to the right until L is reached, when the letter of the message K will be found at the top of the column so located. The novice at this cipher may gain all the necessary practice for its use by enciphering and deciphering the short illustrative example in full.

In the Vigenère cipher it is possible for a given letter to be represented by any letter whatever in cipher; and conversely, any cipher can conceivably be the substitute for any letter whatever in the message.

And this is no doubt responsible for the opinion held by some that this cipher cannot be read without the key. For, it is probably reasoned, how could it be possible to find the meaning of a cipher letter, when that letter can stand for any letter of the alphabet.

A little thought, however, will show that this premise is not altogether true. For in order that a letter in cipher can act as the substitute for all twenty-six letters, would mean also that all twenty-six key-letters would have to be used in enciphering them.

In the ease of any single letter in cipher, it is possible that any key-letter may have been used. But this would not hold for a series of letters, for the reason that the key is the one thing in the Vigenère cipher that remains fixed. And any suppositions as to the identities of letters in cipher must be such as would, at least in some instances, result in repetitions of certain key-letters.

As a matter of fact, a number of methods of deciphering this and similar ciphers have been devised. What is probably the earliest method of solving the Vigenère cipher is described by John Falconer in his cipher book, " Cryptomenysis Patefacta," published at London in 1685.

This method, admittedly fundamental, is nevertheless a very necessary tool in the cryptographer's equipment. For there are numerous instances especially in short messages, of which that about to be deciphered is one, where this method is superior to any other.

Falconer instructs the decipherer first to guess at the identity of short words, obtaining in this way short portions of the key. Fragments of the key obtained by suppositions as to several words, may often be combined, forming a larger portion of the key. Or any part of the key so discovered may be further developed by suppositions as to adjoining letters, either in the message, or in the key itself. The whole number of letters in the key can thus be arrived at, determining the several returns of each alphabet.

To demonstrate this method, it will now be applied to the short illustrative cryptogram just enciphered, which contains one two-letter group, TG; two three-letter groups, AFW and PTD; and three four-letter groups, LIFZ, AWSU, and FJHU.

According to the word frequency table in FLYNN'S for May 16, the most frequently used words of these lengths are those listed in the lines marked (b) of the following tabulation. It is highly probable that one or more of these words, all of them being of high frequency, will occur in the average message. Any additional words desired may also be tried. If some idea is had as to the nature of the message, it is advantageous to try a suspected longer word at the start, since it is often possible to arrive at a large part of the key in this manner immediately.

Each of the cipher groups, in lines (a), is repeated as many times as the number of tentative words for it requires. And beneath each word, in lines (c), are placed the key-letters necessary for deciphering the cipher group into that word.

Thus, if it is assumed that TG=OF , then the key letters used for enciphering the latter must have been FB. This is discovered by tracing down the O-column to the cipher letter T, at the left of which row is found the key-letter F. Similarly, the F-column followed down to G, gives the key-letter B at the left. The individual key for every supposition on the table is obtained in a similar manner.

(a)  Cipher:  TG  TG  TG  TG  TG  TG  TG  TG  TG  TG  TG  TG  TG  TG  etc.
(b)  Message: OF  TO  IN  IT  IS  AS  BE  AT  WE  ON  HE  BY  MY  ME  etc.
(c)  Key:     FB  AS  LT  LN  LO  TO  SC  TN  XC  FT  MC  SI  HI  HC  etc.

(a)  AFW  AFW  AFW  AFW  AFW  AFW  AFW  AFW  AFW  AFW  AFW  AFW  etc.
(b)  THE  AND  YOU  FOR  WAS  NOT  HUT  HIS  ARE  ALL  HAD  HAS  etc.
(c)  HYS  AST  ORG  VRF  EFE  NRD  ZLD  TXE  AOS  AUL  TFT  TFE  etc.

(a)  PTD  PTD  PTD  PTD  PTD  PTD  PTD  PTD  PTD  PTD  PTD  PTD  etc.
(b)  THE  AND  YOU  FOR  WAS  NOT  BUT  HIS  ARE  ALL  HAD  HAS  etc.
(c)  WMZ  PGA  RFJ  KFM  TTL  CFK  OZK  ILL  PCZ  PIS  ITA  ITL  etc.

(a)  FJHU  FJHU  FJHU  FJHU  FJHU  FJHU  FJHU  FJHU  FJHU  FJHU  etc.
(b)  THAT  HAVE  WITH  YOUR  THIS  FROM  THEY  WERE  BEEN  WHEN  etc.
(c)  MCHB  YJMQ  JBON  HVND  MCZC  ASTI  MCDW  JFUG  LFDH  JCDH  etc.

The cipher groups LIFZ and AWSU have not been included in this tabulation for the reason that they did not produce any probable key sequences with any of the common four-letter words.

Examining these lists for tentative keys having letters in common, AS, AST, and ASTI, stick out like so many sore thumbs.

If these supposed parts of the key be now placed above the cipher groups they have deciphered, an interval of 16 letters will be found between AS and AST, and one of 8 letters between AST and ASTI. If these key suppositions are correct, the number of letters in the key must be evenly divisible into both 8 and 16. Keys of 1, 2, or 4 letters are obviously out of the running.

It remains, then, but to try ASTI on the supposition that it is a part of an 8-letter key, with the following result:

                          Interval=16    Interval=8
                    ┌─────────────────┐ ┌────────┐   
Key:     -ast i---- AST i-- --asti ---- AS ti---- ASTI ----as
Cipher:  LIPZ TZKMP AFW PTD JBMABG AWSU TG XANLTF FJHU QCEOCW.
Message: -ing l---- AND h-- --mily ---- TO es---- FROM ----oe.

An examination of the first eight letters of the cipher shows the location of the four discovered key-letters in the whole key—ASTI—to which additional fragments of the key may now be added, if any are available.

The first letter of the fourth cipher group, PTD, has been enciphered with the keyletter I. And the tabulation provides three probable sequences of key-letters for UTD beginning with I, thus:

Key:     AST ILL AST ITA AST ITL
Cipher:  AFW PTD AFW PTD AFW PTD
Message: and his and had and has 

Of these—ASTILL—seems most likely, it being more like an ordinary word, besides and his seems preferable by context in the partly deciphered message. By actual trial —ASTILL— proves workable throughout:

—ING LOU— — AND HIS — —MILY
PL— — TO ESCA— — FROM FR— —CE.

Here —ING is obviously KING, giving the additional key-letter B, which makes PL— — read PL—T (clearly PLOT); FR— —CE becoming FR—NCE (evidently FRANCE) ; thus completing the key, BASTILLE, with the resultant decipherment of the entire message.

In this instance the translation of the message has been reached through the discovery of several parts of the key at different points of the message, the latter having been divided normally into words.

By a simple modification of the method, a similar message, continuously written. could readily be deciphered, even with the discovery of only a single key possibility.

In a message normally divided into words, it was necessary to try probable words only with those cipher groups of the requisite number of letters. In the absence of word divisions, the only difference is that such words must be similarly tried throughout the cryptogram.

To illustrate this, take the above cipher continuously written:

LIFZT ZFMTA FWPTD JBMAE
GAWSU TGXAN LTFFJ HUQCE
OCW.

The trial of any word not in the message will result negatively. The following shows the application of AND, revealing (2 ) ISW, and (10) AST, as possible key fragments.

Having found supposed parts of the key, the problem is now to find the whole number of letters in it. And this is done by applying the tentative group as a key throughout the cryptogram, searching for those points at equal intervals where normal sequences result in the partly deciphered message.

ISW, tried in this way, can be rejected. But AST (the key by which AFW=AND) fits the lock:

        1    2    3    4    5    6    7    8    9    (10) 11   12   13
Key:    AST  AST  AST  AST  AST  AST  AST  AST  AST  AST  AST  AST  AST
Cipher: LIF  IFZ  FZT  ZTZ  TZF  ZFM  FMT  MTA  TAF  AFW  FWP  WPT  PTD
Text:   1qm  ING  fha  zbg  thm  znt  fua  mbh  tim  AND  few  wxa  pbk 

        14   15   16   17   18
        AST  AST  AST  AST  AST  etc.
        TDJ  DJB  JBM  BMA  MAE  etc.
        tlq  dri  jjt  buh  MIL  etc.

Here a key of eight letters is indicated, since the normal sequences (2) ING, (10) AND, (18) MIL, occur at intervals of eight.

Applying AST as part of the supposed eight-letter key, the cryptogram will now read:

Key:    —AST—————AST—————AST—————AST—————AST—————AS.
Cipher: LIFZTZFMTAFWPTDJBMAEGAWSUTGXANLTFFJHUQCEOCW
Text:   —ING—————AND—————MIL—————TOE—————FRO—————CE  

By developing the key and the message in this instance, as before, the whole key and entire message may be obtained.

As already mentioned, the Vigenère cipher was used by the Confederates during the Civil War. They thought, of course, that it was absolutely safe. But such of their messages as were captured were deciphered by the Federal cipher operators by guessing at the meanings of words substantially according to the method outlined by Falconer nearly two centuries previously.

In some instances normal word divisions were retained in their cipher, and in other cases the writing was continuous. Only a few keys seem to have been used, and apparently no need was felt of changing them frequently, for they are known to have been used over long periods of time.

         1    2    3    4    5    6    7    8    9    10   11   12
Text:   and  and  and  and  and  and  and  and  and  and  and  and  etc.
Cipher: LIF  IFZ  FZT  ZTZ  TZF  ZFM  FMT  MTA  TAF  AFW  FWP  WPT  etc.
Key:  LVC  ISW  FMQ  ZGW  TMC  ZSI  FZQ  MGX  TNC  AST  FJM  WCQ  etc.    

It was a common practice with the Confederates to encipher only the more significant words of their messages, leaving the rest in plain English, as in Cipher No. 1 below. This only made an insecure cipher still less safe, for words in plain text in any cipher often suggest the meanings of those that may have been enciphered.

THE CONFEDERATE CIPHERS

Two of these captured Confederate messages are subjoined. Which will give the reader a chance to match wits with the Federal operators who actually solved them in a time of need.

And this process should be interesting from more than a single point of view.

For not only will the successful decipherer unravel two messages of historic importance, but he will also discover two of the closely guarded keys actually employed by the Confederates.

And besides he will find for himself that this much vaunted undecipherable cipher is really decipherable after all!

CIPHER No. 1 (Vigenère chiffre carré).

I recommend that the TSVSMEE FN QOUTWP RFATVVMP UBWAQBRTM EXFVXJ and ISWAQJRU KTMTL are not of immediate necessity, UV KPGFMBPGR MPC THNLFL should be LMQHTSP.

CIPHER No. 2 (Vigenère chiffre carré).

VVQ ECILMYMPM RVCOG UI
LHOMNIDES KFCH KDF WASPTF
US TFCFSTO ABXC BJX AZJKHMGJ
SIIMIVBCFQ QB NDEL UEISU HT KFG
AUHD EGH OPCM MFS UVAJWH
XRYMCOCI YU DDDXTMPT IU
ICJQKPXT ES VVJAU MVRR
TWHTC ABXC IU EOIEG O RDCGX
EN UCR PV NTIPTYXEC RQVARIYYB
RGZQ RSPZ RKSJCPH PTAX RSP
EKEZ RAECDSTRZPT MZMSEB ACGG
NSFQVVF MC KFG SMHE FTRF WH
MVV KKGE PYH FEFM CKFRLI-
SYTYXL XJ JTTBX RQ HTXD
WBHZ AWVV FD ACGG AVXWZVV
YCIAG OE NZY FFT LGXA SCUH.

READER TO READER !

That the Vigenère chiffre carré still manages to cling to its old reputation of being undecipherable without the key, is no doubt because information about cryptography is neither widely spread nor readily accessible.

This can be no discredit to the individual, however, when even governments, and in comparatively recent times, have seen fit to officially sanction its use.

At any rate, a number of messages in this cipher, a few of which follow, have been submitted to this department by readers who do not hesitate to say that their cryptograms are impossible of solution without the keys.

Nos. 3 and 5 are straight Vigenère ciphers. In No. 4 the method of using the alphabetic square has been slightly modified, but not to an extent that will prevent its being solved by the present method.

No. 6 combines in an ingenious way the Vigenère block alphabet with an early system of filing finger-prints under the Henry classification.

Give these a trial, and send in your own Vigenère ciphers, using your own key, for your fellow readers to ponder over.

DEAR SIR:

I have been reading FLYNN'S for some time, and have also read a few things that have been in your department under the heading of " Solving Cipher Secrets."

About five years ago I met a young fellow who was interested in the woods, and we paired off. We have been pards ever since. It was on one of our trips that he told me about a cipher that he had.

Perk up your cars for this is a good one. Unless you had the key word, and, of course, this is where I want to keep you guessing, it was impossible to solve it.

Any word that you can think of can be a key word. The way he tells me, this is an old cipher, and was used many years ago.. Where he got it he doesn't remember. By the way, he is sitting here reading this over my shoulder.

We are both in on this, so we are going to make it as interesting as possible. Even though the message is short it doesn't deal with buried treasure. To us it has been buried treasure saved for the end of our tramps.

F. B. WILLITS.

Harrisburg, Pa.

A. J. LINDSAY.

CIPHER No. 3 (F. B. Willits-A. J. Lindsay).

Here is the message to FLYNN'S from two who have found it to he a boon companion on trips to the mountains:

KVVWAAK MLQ ORWS DRBJWI TL
BHJ ULAXF SX CCYVYNZSFYQ FD
MHE HNACF NFI EMONUHZ.

After you have found the apparent key in No. 4, it should be interesting to work from it to the real key, thus discovering the modification above mentioned responsible for the difference.

DEAR SIR:

Will you please decipher the following message?

1 think it is indecipherable. I hope to find out if I am right or wrong in the near future in FI.YNN'S cipher department.

A J. SIMON.

Brooklyn, N. Y.

CIPHER No. 4 (A. J. Simon).

  UZDCB'X XG TCS TU HMT PJHH
BTSPAM RPUFOWSTG NC BJL MTGY
HXHD.

The next cipher, submitted by Ralph Raphael, Worcester, Massachusetts, may prove a little more difficult of solution by the method described in this issue, for the reason that the key is rather long, and common words rather scarce in the message.

CIPHER No. 5 (Ralph Raphael).

YHRLM  IMKTG  PTEXE  FMZPW  RECME  WGCFC
HCNSH  VCAJE  OATSS  ABRVQ  QWUAT  CMIMK
TGPUE  OCRPH  UABWJ  SWEVI  GYBXK  OJOSK
SWRFC  ZZKCX  ZXTMP  GGMHB  AGPAJ  QGVNB
UWFDH  CTEZI  CSLMB  YYCZY  AAVM.

The key to Cipher No. 6 was designed by W. W. Reeves, F. P. F. , of San Francisco, California. Mr. Reeves does not claim any originality in his cipher merely the adaptation of an idea.

CIPHER No. 6 (W. W. Reeves, F. P. E.).

13 5 6 11   4 8   2   15 8 13 14
—— — — ——   — —   —   —— — —— ——
8  4 4 9    6 12  26  15 8 18 18

14 12 9  5 3 6 3 5   7 19 9  6 8
—— —— —  — — — — —   — —— —  — —
17 19 11 5 5 9 3 26  9 22 12 3 4

12 7  17 9  17   18 22 22 11
—— —— —— —— ——   —— —— —— ——
23 24 24 10 10   15 14 17 21

In Cipher No. 7, by M. Walker, Akron, Ohio, the Vigenère table has been used exactly as it was in No. 4 above. Besides, Mr. Walker has used a continuous non-repeating key, thus avoiding the repetition of a single key-word or key-phrase. The first word of Mr. Walker's key is one of the most used words in the English language.

CIPHER No. 7 (M. Walker).

CPTBT  LLNYN  MLFNB  TJFNX
RBIUE  CCLFU  WMRZF  UONZN
VHOAF  NSSJD  VFWAZ  TZHRF
BHBSW  XIQPN  TGTOZ  NLGRX
WX.

If you still have your copy of FLYNN'S for October 31, with the method described in this issue you should be able to solve No. 5 (Wm. E. Bowns)—a straight Vigenère—in short order. It may be said here that any previous issue of FLYNN' S can be had postpaid for ten cents.

The solutions to all of the ciphers in this issue, including No. 5 of October 31 just mentioned, will be given in next Solving Cipher Secrets.

Submit your solutions, and compare your score with that of others.

ANSWERS TO JANUARY 23 CIPHERS

Cipher No. 1 (W. B. Tyler to F. A. Poe)

The letters of each word were also transposed by being written in reverse order, in the January 23 issue of FLYNN'S, was thus:

Cipher:       , † § : ‡ ] [ , ? ‡ ) , [   etc.
Substituting: E H T L U O S E R U C E S &mdash etc.
Transposing:  The   soul    secure

enciphered in the following simple substitution alphabet of typographical characters:

The text is a quotation from Act V, Scene I, of Joseph Addison's " Cato ": "The soul, secure in her existence," et cetera.

In Cipher No. 2 each figure could be the substitute for three different letters, depending on whether it was followed by a dash, another figure, or a space, according to the following key:

1 2 3 4 5 6 7 8 9
A B C D E F G H I (dash)
J K L M N O P Q R (figure)
S T U V W X Y Z — (space)

Here is a small section of this message deciphered:

Cipher:        9—5765—1 4—1—7 1—54—5—4  etc.
Substituting:  I NPOE S D A Y A ND E V  etc.
Spacing:       In Poe's day and ev-     etc.
    

The answer to No. 3 (Hobart Hollis), the ten dollar prize cipher, will be printed in the next issue of this department. You can't afford to miss it.

In No. 4 (D. Washburn Hall) any letter is represented in cipher by the sixth letter following it in alphabetical order, G being thus used for A, H for B, et cetera, the entire message having also been written backward.

The solution to No. 5 (Mrs. A. J. Hyatt) is:- " FLYNN'S magazine gives me many hours of enjoyment, and the cipher department is so good that it robs me of my beauty sleep."

To decipher, read the numbers downward by columns, and divide each number by 25, disregarding any remainders. The quotients so obtained will then represent letters from the simple numerical alphabet:

        (space)=27 ; A=26; B=25; . . . . Z=1.

No. 6 (William H. B. Woodbury, F. P. E.) employed the following reciprocal key, in which any letter is the substitute for that with which it is paired: A=B ; B=A ;E=C ; C=E ; et cetera.

A E I O U L M N R W X Y Z
B C D F G H J K P Q S T V 

His message was: " What shall it profit a man if he construct ciphers just to have some one solve them before the ink is dry? Have never missed a number of FLYNN'S since first published."

The complete translation to No. 7 (J. W. B.) explains the system used:

"This cipher is called the D. A. Code, or the Double and Add system. Double the first figure of any pair, and add the second to get the number of the letter in the alphabet. At the end of a word the first figure of the next pair is repeated."

If you will refer to FLYNN'S for January 23 you will note that J. W. B. very cleverly expressed his key in the italicized words of the last paragraph of his letter. This ingenious key uses every combination of two figures from 01 to 98 inclusive, several substitutes thus being available for some of the letters.

But the simple formula, "double the first figure and add the second," makes it possible to do all the work mentally. Thus 84, the first two figures of the cipher in question, stands for T, since 8 doubled, plus 4, equals 20, the position occupied by T in the alphabet.

THE ROLL OF HONOR

Charles P. Winsor, of Boston, Massachusetts, leads the list of December 5 cipher solvers, with correct solutions to four ciphers. No solutions were submitted to ciphers Nos. 3, 4, and 7.

• Charles P. Winsor, Boston, Massachusetts. (1—2—5—6.)
• Arthur Bellamy, Boston, Massachusetts. (1—2.)
• G. A. Ferrcll, Bessemer, Alabama, (1—2.)
• Charles C. Fulton, Omaha, Nebraska. (1—2.)
• A. P. Schmutz, Philadelphia, Pennsylvania. (1—2.)

All of these correspondents accompanied their solutions with interesting explanations of the methods used, which unfortunately cannot be repeated here for lack of space.