Better word search building - Filling Space

Question

I have been unsatisfied with the quality of word searches I am producing.

I've been filling the squares that are empty after the words are placed with random letters.

Too many of the empty squares are getting filled with q,z, and other uncommon letters, making it too easy.

I tried using only random letters that occur in the clue words but that ended up with small words dotted all over when the list contained lots of short similar word.

I could test each random letter to check if it makes a word by mistake but that will only rule out clue words unless I start using a dictionary which I had hoped to avoid.

As their are only $26^{2}$ pairs of characters I was wondering if any body could list the most and least common as this would help me pick my random letters.

TL:DR

Any suggestion on how to pick the random letters that fill up space in a word search? I was thinking a making pairs of either the least or most common pairs of English letters.

Answer 1

Using the Enable word list, the least common pairs, each occurring only once, are:

fk - offkey
jk - pirojki
qe - sheqel

Edit: Pairs that appear twice (although you could probably call these pairs that appear once, as well, due to plurals):

wj - blowjob
hv - boshvark
cw - cwm
gv - dogvane
vz - evzone
zg - fizgig
zt - gewurztraminer
vd - havdalah
zj - muzjik
xv - poxvirus
qo - qoph
qw - qwerty
js - rijsttafel
zd - samizdat
yy - sayyid
cp - secpar
xd - sexdecillion
jd - slojd

Pairs that appear three times:

vk - sovkhoz, sovkhozes, sovkhozy
bz - subzero, subzone, subzones

Answer 2

Using the word list provided by Golden Dragon here are the frequencies for every pair of characters.

The word list had 172820 words including plurals.

There were 84 pairs that weren't found in any word.

cg,qm,jr,jq,zr,kq,gx,tx,ql,mx,jp,kx,gq,zf,kz,vb,qt,qk,jm,jl,wv,vc,jt,jh,vf,vh,jg,jf,qv,bq,yq,jc,jb,qj,qx,bx,vj,fz,fx,fv,vm,qy,cf,qn,qz,xz,cj,rx,xx,fq,jv,vp,vq,wx,jw,xr,cv,cx,vt,qh,qg,jx,jy,zx,vw,vx,jz,xk,pq,qf,xj,pv,px,qr,qp,sx,qq,pz,hx,dx,qd,qc,wq,qb

Below are plots of the pair frequencies. A is top left, Z is bottom right.

Most common is white:

Those that occur more than 512 times or more are pure white in this one:

Because of what I had hoped to do with this data regarding word searches, think the pairs need to be treated as reversible, and to list of pairs that occur 0 times needs to extended to include runs of 3 and 4 letters. Here is the data in machine and human readable formats:

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0 - zf kq jl zx qp fv jz qj kx qy jh xx jp vt vh cv qn vq xz wx vc jt xk qt jm bq jb tx wq qz zr hx vm jw jg vb pq qm sx gq jv qr qf ql gx qc qx cj mx jr qd qg px jf qv vw qk fz bx qq qh cf jx xr cg pv vj vx cx xj vp jy fx qb wv yq fq kz jc jq rx dx pz vf 
1 - qe jk fk 
2 - xv zt jd zd qw js yy wj xd hv cw zg vd qo vz gv cp zj 
3 - bz vk 
4 - zk mq zb gj jj cb fg fj xg jn zc lj zq zs gz xn 
5 - qs xq sz lx vg 
6 - vn gk hz tq mz mj mg hq 
7 - qa zn zv 
8 - fw vl xm dq zh hj uw 
9 - zw wz qi 
10 - zp fc mk zm yj cm 
11 - kj dk 
12 - lq fm bk xb fd pj 
13 - fp 
14 - md gc uq 
15 - xw 
16 - lz 
17 - wg vs 
18 - yv xf 
20 - kv kg 
21 - cz 
22 - tk 
23 - hg fh 
24 - fn gp 
26 - vv fb ww 
27 - dz wc bg 
28 - tj iy pk hk 
29 - yk uh 
30 - uu cd xl mt 
32 - mh 
33 - uj 
34 - tv vr 
36 - mv kc 
38 - kd mw 
41 - pg xs 
42 - wu bw 
43 - pd 
45 - hh 
46 - hc mc 
47 - ij nx kk sj 
48 - sv 
49 - bf 
50 - gd zu 
51 - cn 
52 - bp 
53 - uy 
54 - rz 
55 - ii hd mr 
57 - bh 
58 - vy iw yx 
59 - wp gf 
65 - pc 
66 - kp dt rj 
67 - wm bv rq 
68 - wy 
69 - bn 
71 - lr 
72 - pf 
74 - yu wf 
75 - gw yh 
76 - aa yf 
77 - pw 
78 - hp 
79 - wt 
80 - gt 
81 - oj 
88 - pm cq 
89 - bm 
93 - pb 
96 - wk 
97 - kf 
100 - xh gb kt 
101 - km hf 
103 - ao 
106 - mf 
107 - yz 
113 - lw dc 
119 - sg 
121 - dp dj 
122 - aq ih 
124 - aj 
125 - bj 
129 - oq 
130 - bc 
131 - kw 
137 - sd lh 
139 - bd pn 
145 - uz 
146 - yw 
149 - td 
150 - ku 
152 - ux 
154 - wd wb tg 
156 - df 
158 - kr 
160 - xu 
161 - kb 
162 - hw hb 
165 - zl 
170 - kh 
171 - nz nq 
172 - ej 
174 - ml sr 
182 - zy 
183 - dv 
190 - uv 
196 - dh 
206 - ez 
207 - ji 
209 - tp 
210 - nj 
213 - bt 
215 - uk 
223 - vu 
224 - tz 
227 - db 
230 - yb 
239 - xy 
242 - iq by 
244 - yg 
253 - sb 
262 - dw 
264 - sf 
269 - lg 
273 - lb 
275 - dm 
278 - xc xo 
308 - oh 
310 - fs 
314 - gm 
316 - ek 
317 - mn 
323 - tb 
324 - nw 
331 - oz 
332 - ky 
339 - ix 
346 - wl zz 
349 - xa 
353 - ko 
354 - tn 
355 - rw 
360 - hm 
369 - tm 
387 - tf 
397 - wr 
403 - kn 
409 - sq 
424 - fy 
426 - eq 
428 - lp 
429 - lv 
436 - uo 
438 - np 
441 - nm 
451 - nb 
452 - lf 
467 - hn 
471 - yd 
482 - nr 
483 - nh 
488 - lc 
508 - jo 
511 - ah 
516 - uf 
518 - lk 
526 - ln 
527 - lm 
528 - je 
537 - oy 
539 - xp 
543 - tw 
549 - ax 
561 - py 
562 - ws 
563 - xt 
574 - dg 
584 - rh 
587 - ny 
589 - ja 
593 - nl 
602 - yo 
608 - kl wn 
619 - dy 
625 - zo 
626 - yr 
627 - dn 
628 - eh 
629 - hl 
645 - ju 
654 - gy 
672 - ft 
679 - iu 
686 - xe 
701 - ya 
705 - az 
709 - rf 
724 - yn 
737 - my 
758 - wh 
760 - sw 
769 - rv 
771 - cs 
773 - yi 
780 - nv 
785 - hs 
806 - yt 
841 - ox 
863 - yc 
867 - ik ye 
876 - bs 
887 - ka 
891 - ey 
903 - ae 
905 - ym 
917 - xi 
940 - ok 
946 - sk 
954 - bb 
959 - of 
963 - cc 
968 - dd 
1001 - af 
1011 - tc 
1071 - aw 
1113 - yl 
1117 - rk 
1123 - ld 
1127 - cy 
1140 - oe 
1141 - ht 
1152 - hu 
1165 - gn 
1178 - vo 
1179 - rl 
1195 - ug 
1203 - nk 
1221 - sy 
1249 - gg 
1257 - ew 
1265 - eu 
1270 - ak 
1275 - eb 
1291 - yp 
1319 - nf 
1329 - zi 
1343 - za 
1347 - fr 
1372 - ay 
1393 - hr dl 
1408 - rp 
1411 - ud 
1439 - nu 
1463 - rb 
1498 - rg 
1516 - ys 
1560 - gh 
1612 - ks 
1630 - tl 
1633 - sn 
1634 - du 
1637 - wo 
1651 - ue 
1652 - av 
1656 - ev 
1660 - gu 
1661 - ff 
1694 - pt 
1695 - nn pu 
1723 - oa 
1763 - sl 
1765 - eo 
1766 - ei 
1776 - gs 
1784 - wi 
1793 - mm 
1841 - ui 
1843 - fa 
1857 - we 
1862 - ob ua 
1870 - pp 
1876 - lt 
1880 - rn 
1903 - go 
1934 - fu 
1943 - mu 
1948 - ps 
1959 - ef 
1985 - ib 
2026 - eg 
2029 - dr 
2042 - mb 
2090 - ub 
2135 - uc 
2139 - fl 
2187 - au 
2195 - bu 
2234 - oi 
2244 - rc 
2283 - gl 
2294 - up 
2343 - va 
2387 - hy 
2392 - fo 
2421 - ms 
2439 - cl 
2449 - ry wa 
2455 - fe 
2474 - rr 
2483 - ex 
2506 - qu 
2511 - ow 
2526 - ki 
2528 - br 
2533 - if 
2544 - rd 
2613 - ty 
2629 - ip 
2633 - rm 
2707 - ls 
2718 - ds 
2722 - da 
2866 - ru 
2892 - ov 
2903 - od 
3049 - sm 
3053 - tt 
3062 - lu 
3168 - do 
3194 - tu 
3203 - ga 
3243 - ze 
3246 - cu 
3250 - ai 
3297 - ig 
3312 - vi 
3350 - ir 
3368 - mp 
3397 - pl 
3542 - ep 
3558 - bo 
3572 - um 
3601 - gr 
3675 - fi 
3710 - ag 
3760 - sa 
3762 - so rt 
3773 - cr 
3815 - iv 
3822 - oo 
3829 - sp 
3842 - oc 
3861 - ba 
3866 - ke 
3878 - ee 
3904 - ck 
3942 - gi 
3972 - be 
3978 - sc 
4078 - im 
4212 - su 
4272 - bi 
4367 - ap 
4454 - ad 
4522 - iz 
4523 - pi 
4668 - og 
4694 - ct 
4829 - am 
4897 - mo 
4905 - ut 
4920 - id 
4938 - op 
4966 - pa 
4971 - em 
4999 - ph 
5060 - bl 
5174 - ot 
5254 - ci 
5259 - po 
5413 - ab 
5442 - ha 
5513 - sh 
5616 - ge 
5757 - hi 
5772 - ul 
5781 - nc 
5893 - ia 
5909 - pr 
6010 - no 
6031 - th 
6045 - ho 
6078 - os 
6089 - ce 
6130 - ec 
6194 - ur 
6288 - ts 
6346 - nd 
6400 - om 
6578 - us 
6754 - ea 
6798 - ac 
7042 - na 
7411 - ma 
7574 - si 
7654 - il 
7707 - ly 
7710 - pe 
7789 - et 
7805 - un 
7821 - ch 
7906 - ol 
7919 - lo 
7986 - ou 
8017 - el 
8085 - mi 
8129 - ve 
8215 - ni 
8258 - to 
8279 - he 
8313 - ll 
8333 - me 
8437 - as 
8508 - di 
8555 - tr 
8619 - ta 
9217 - ca 
9860 - rs 
10163 - ns 
10367 - ie 
10372 - la 
10554 - de 
10626 - io 
10884 - co 
11034 - ss 
11172 - it 
11703 - ro 
12409 - or 
12621 - nt 
12946 - ic 
13056 - ar 
13111 - se 
14857 - an 
14947 - ra 
15027 - ri 
15180 - ne 
15239 - li 
15383 - le 
15552 - is 
15761 - al 
16378 - st 
16978 - en 
17018 - ed 
17683 - ng 
19348 - at 
19890 - on 
19901 - re 
20028 - te 
24411 - ti 
32289 - er 
32894 - in 
38404 - es