If you're a fan (or addict) of the puzzles once known as Cross Sums, and now known as Kakuro, you may have seen the long charts that are made up for all the possible combinations adding up to each number from 3 to 45. They fill pages. And all the information on them that you need to know can be condensed onto a small card that you can carry in your wallet, or draw for yourself on the corner of the page before you start working a puzzle. In fact, it's so small, I normally add a help for cryptograms as well, which gives the name to the cards: ETAOIN SHRDLU (which I pronounce Ee'tay-oh-in 'Sherd-lu). More on the cryptogram key below.

Using the Kakuro Kracker table: This table can be either vertical or horizontal, but to keep it simple I'll refer only to the vertical version directly above, and not the horizontal one at the top of the page. The two center columns are the summations, the outer columns are the enumerations. To find the numbers that can be added to a sum, step down the number of lines that you have squares to fill in your puzzle. For example, if you have 4 squares that are supposed to total to 10, look in the 4th row down, and in the second column you have 10, indicating that all four of the numbers from 1 to 4 are needed, and none of those from 5 to 9 are permitted. Likewise, if the sum you had been aiming for was 30, the 30 in the third column indicates that the 9 to 6 in the right-hand column are all required, and the numbers below that, 5 to 1, are prohibited. This also tells you that 4 digits can not add up to anything higher than 30 or lower than 10, sometimes a useful fact if you're trying to deduce the remaining 4 digits of a 7 digit series.

But what if the sum is somewhere between 10 and 30? Then we need to extrapolate just a little. Imagine the numbers in the center columns counting onward from the 10 or 30 marks, as shown in red in the illustration at left. 11 is now opposite the 5, and the only 4 digits that add up to 11 are 1+2+3+5.

Continuing on, the 12 is opposite the 6, and there are two combinations that total to 12, 1+2+3+6, or 1+2+4+5; so the card tells us that 6 is the highest digit that can appear. Also, counting in the other direction, shown this time by blue numbers, the 12 is opposite the 2, and we notice that both the allowable combinations require all the digits from 1 to 2.

Following this pattern, we can see that for sums of 10, 11, 12, 13, 14, 26, 27, 28, 29, & 30, we can gain some useful information. But at 15 and 25, and any number between them, the information is that any number from 1 to 9 is possible, and no number is required. So we'd best do our prospecting in another part of the puzzle if we can, until some crossing summations narrow our choices a bit more.

As you use this table, you'll find it tells more information. For instance in the example we've been using, 20 is midway between the possible sums for 4 numbers of 10 to 30. So a number like 16, which is less than 20 (and closer to 10) will be narrowed down by finding large numbers, such as 8 or 9, in it, but small numbers like 1 or 2 are to be expected, and so won't help as much in identifying the remaining digits. I've been using this since I first developed it years ago, and I'm still finding new secrets in it.

Creating the table: If you will print out just the first sheet of this webpage, on cardstock if available, the two cards at the top of the page will print out much clearer than the slightly distorted versions which appear on this page, and can be cut into convenient wallet size cards. For those times when you don't have a card with you, it's easy enough to draw a copy on the corner of your worksheet. The two outer columns are simple, just 1 to 9 and 9 to 1. The inner columns begin with the same number as the corresponding outer columns, and add the next number in line for each. Thus 9, 9 (the previous number brought forward) + 8 (the next number) = 17, 17+7=24, 24+6=30, 30+5=35, and so on. For the other column, 1, 1 (the previous number brought forward) + 2 (the next number) = 3, 3+3=6, 6+4=10, and so on. A line to divide each group of three rows (or columns, if you prefer to do it horizontally) is helpful. They tend to get out of alignment (making them hard to read) unless the 1-9 column is done first, then the guide lines drawn, and finally the other three columns filled in.

Using the Cryptogram Key: I'll assume that you already have a rudimentary idea of how to solve a cryptogram (there are plenty of other places to get that information). There are three categories of cryptograms used as puzzles:

• lists of associated terms (sometimes called crypto-lists or crypto-families),
• quotations (usually known as crypto-quotes, and usually containing the author's name within the encyphering), and
• phrases made up specifically for cryptogramizing (generally known as crypto-quips).
  
  In collections such as puzzle magazines, -quotes and -quips are often intermixed under either title.

Cryptolists present a slightly different challenge from quotes or quips because they don't contain the small common words (articles, pronouns, prepositions) that often provide keys to the more specific words in a sentence. This also skews the letter frequency, since the most common word in English is "the", and the most common letter combination is "th", used not only in the but also this, them, these, those, etc. Also, most sentences have at least one word with an "s" added on to the end, either nouns made plural, or verbs that go with singular nouns. The distribution of Scrabble tiles gives a good indicator of letter frequencies for cryptolist puzzles. There is also a type of crossword sometimes called codeword which also uses letter frequency as a solution method, and it has the same limitation as cryptolists.

Sometimes cryptoquips are also constructed so as to skew frequency distributions of letters, but this is harder to do, and doesn't upset it as much. The simplest way to do this is to use words with lots of odd letters, like "Quizzes equal inquiries concerning puzzles." More interesting cryptoquips may be composed as lipograms, purposely avoiding some common letter(s), or as pangrams (often misspelt as panagram), sentences containing every letter of the alphabet, or even double pangrams, paragraphs containing each letter at least twice.

With these factors in mind, the sequence of letter frequency for a normal, untweaked sentence in modern conversational English begins E, T, A, O, I, N, S, H, R, D, L, U. (Keep in mind that T, S, & H may all move down a position or two for a cryptolist or codeword puzzle; E is so much more frequent than anything else that it remains first.) (As mentioned in the top half of this page, ETAOIN SHRDLU is pronounce Ee'tay-oh-in 'Sherd-lu). Beyond that, the frequencies get so small that assigning them a fixed order gets somewhat subjective. C, M are cited as the next most common, F, P, Y, W, G, B, V in something like that order are next, and K, J, X, Q, & Z are the rarest.

The subject matter, level of writing, and even the date as our language evolves all affect letter frequency. Several sources were consulted in coming up with the ordering on this card. You probably know the Qwerty keyboard was designed to slow down typist, but the typewriter wasn't the only machine designed to produce writing. The Linotype machine for typesetting was developed independently, and its keyboard was designed for efficiency. Several companies manufactured typesetting machines, and the keyboards differed, sometimes markedly, but the positions of at least the first 12 most-used keys were universal. This picture, from http://www.hpricecpa.com/typewriters.html, is the clearest picture of a linotype keyboard that I've only been able to find on the internet, and it's of an early typewriter with a linotype-style key layout. The white keys are lower-case, the black ones uppercase, numbers, and most punctuation marks. The 5 orange keys may have been shift or control keys, and the long keys at the bottom probably space and backspace. Note that there are no 0, 1, or ! keys.

click images for larger versions

This keyboard above gives the sequence (translated horizontally) as

ETAOIN SHRDLU
CMFWYP VBGKQJ XZ

According to The Straight Dope and other sources, the full sequence was

ETAOIN SHRDLU
CMFGYP WBVKXJ QZ

A book on cypers (long since lost, so citation not available) gave the letter frequency sequence as

ETAOIN SHRDLU
CMPFY WGBVK JXZQ

Amalgamating all these together gave the sequence used on the card. This is for 20th century English. It would probably be somewhat different for, say, the 16th century English of Shakespeare's writing, or for Spanish or German or any other language. In French, the letter frequency sequence starts 'ESARIN TULOM', and French linotype machines had the leftmost keys as 'ELAOIN SDRÉTU'.

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