5-Cubes & Dice

7/21/2019 5-Cubes & Dice

1/18

M@BQKJ KINJNQ^

Anh`

Anh` ns k tcr`` anm`bsndbkj fngur` wntc kjj df nts

surfkh`s bumi`rà.

Nmpdrtkbt pdnbts kidut Anh` 5

(n) K anh` cks snx surfkh`s kba kjj df tc`m kr`

bumi`rà frdm ; td 6.

(nn) Nf tc` surfkh`s df anh` kr` ubfdjaà kba pjkhà

db k pjkb`, tc` fngur` df anh` sd ditknbà wnjj

jddl jnl` db` df tc` fdjjdwnbg fngur`s.

C`r`, nb tc` fngur` ; w` fnba tckt

;

9 8 1

0

6

Bumi`r ; ns dppdsnt` td 0

Bumi`r 9 ns dppdsnt` td 1

Bumi`r 8 ns dppdsnt` td 6.

C`r` nb tc` fngur` 9 w` fnba tckt

; 9

8

1

0 6




C`r` nb tc` fngur` 8 w` fnba tckt

;

9

8 1

0

6




(nnn) K bumi`r db db` surfkh` df anh` cks db`

bumi`rs dppdsnt` td nt kba fdur anff`r`bt bumi`rs

db nts kaekh`bt surfkh`s.

[us stndb db anh` ckv` i``b hjkssnfnà uba r tcr``

anff`r`bt hkt`gdr ns. Nb tc` fdjjdwnbg pkrkgrkpcsanff`r`bt typ`s df qu`stndbs ckv` i``b anshussà wntc

tc` c`jp df `xkmpj`s uba`r anff`r`bt hkt`gdrn`s.

@SKM_J@XHkt`gdry N

@x.; K anh` cks i``b tcrdwb fdur tnm`s kba prdauh`s

fdjjdwnbg r`sujts.

8 9 0 8

6 0 8 9

; 1 ; 6

(n) (nn) (nnn) (nv)

Wcnhc bumi`r wnjj kpp`kr dppdsnt` td tc`

bumi`r 8 4

(K) 1 (I) 0 (H) 6 (A) ;

Xdj. Frdm tc` fngur`s (n), (nn) kba (nv) w` fnba tckt

bumi`rs 6, ;, 0 kba 9 kpp`kr db tc` kaekh`bt

surfkh`s td tc` bumi`r 8. Qc`r`fdr`, bumi`r 1

wnjj i` dppdsnt` td bumi`r 8. C`bh` dptndb (k) ns

tc` kbsw`r.

Hkt`gdry NN

@x. 9 Qc` fngur`s gnv`b i`jdw scdw tc` twd anff`r`btpdsntndbs df k anh`. Wcnhc bumi`r wnjj kpp`kr td

bumi`r 9.

(K) 8 (I) 1 (H) 0 (A) 6

8 1

1 0

9 6

(n) (nn)

0HCK_Q@Y

HP @X ANH@

7/21/2019 5-Cubes & Dice

2/18

M@BQKJ KINJNQ^

Xdj. Qc` kidv` qu stndb, wc r` dbjy twd pdsntndbs

df k anh` kr` gnv`b, hkb `ksnjy i` sdjvà wntc tc`

fdjjdwnbg m`tcda.

Xt`p N 5 Qc` anh`, wc`b ubfdjaà, wnjj kpp`kr ks scdwb

nb tc` fngur` gnv`b db tc` rngct sna`.Xt`p NN5 Wrnt` tc` hdmmdb bumi`r td idtc tc` anh` nb tc`

mnaaj` ijdhl. Xnbh` hdmmdb bumi`r ns 1, c`bh`

bumi`r 1 wnjj kpp`kr nb tc` h`btrkj ijdhl.

Xt p NNN5 Hdbsna`r tc` fngur` (n) kba wrnt` tc` fnrst bumi`r

nb tc` kbtn-hjdhlwns` anr`htndb df bumi`r 1,

(hdmmdb bumi`r) nb ijdhl N kba s`hdba bumi`r

nb ijdhl NN. Qc`r`fdr`, bumi`rs 8 kba 9 i`nbg tc`

fnrst kba s`hdba bumi`r td 1 nb kbtnhjdhlwns

anr`htndbs r`sp`htnv`jy, wnjj kpp`kr nb ijdhl N &

NN r`sp`htnv`jy.

NN

9

NNN 6 1 8 N

0 NV

;

Xt`pNV5 Hdbsna`r fng (nn) kba wrnt` fnrst kba s`hdba

bumi`r nb tc` kbtnhjdhlwns` anr`htndb td bumi`r

1, (hdmmdb bumi`r) nb ijdhl (nnn) & (nv) C`bh`

bumi`r 6 kba 0 wnjj kpp`kr nb tc` ijdhls NNN kba

NV r`sp`htnv`jy.

Xt`pV5 Wrnt` r`mknbnbg bumi`r nb tc` r`mknbnbg ijdhl.

Qc`r`fdr`, bumi`r ; wnjj hdm` nb tc` r`mknbnbg

ijdhl.

Bdw, frdm tc` ubfdjaà fngur`s w` fnba tckt bumi`r

dppdsnt` td 6 ns 8, bumi`r dppdsnt` td 9 ns 0 kba

bumi`r dppdsnt` td 1 ns ;. Qc`r`fdr`, dptndb (h) ns

dur kbsw`r.

Hkt`gdry NNN

@x.8 Frdm tc` fdjjdwnbg fngur`s df anh`. Fnba wcnhc

bumi`r wnjj hdm` nb pjkh` df 4

9 ; 6

1 0 ;

8 9 4

(n) (nn) (nnn)

(K) 1 (I) 0 (H) 9 (A) 8

Xdj. Nf tc` kidv` anh` ns ubfdjaà, nt wnjj jddl jnl` ks

tc` fng gnv`b i`jdw. (Xtua`bt scduja fdjjdw tc`

m`tcdas ks `xpjknbà nb tc` pr`vndus `xkmpj` td

fnba tc` kpprdprnkt` pjkh` df tc` bumi`rs

kpp`krnbg db tc` anff`r`bt surfkh`s df tc` anh` nbtc` fngur`).

1

0 9 8

;

6

Bdw tc` bumi`r nb pjkh` df '4' hkb i` ditknbà

iy mklnbg k sjngct hckbg` nb tc` fngur` ks gnv`b

c`r`.

Bdw hdmpkrnbg fng (nnn) ks kidv`, w` g`t tckt

bumi`r nb pjkh` df 4 ns 8

1

0 9

; 8

6

Hkt`gdry NV

@x.1 Wcnhc df tc` fdjjdwnbg anh`s ns na`btnhkj td tc`

ubfdjaà fngur` ks scdws c`r` 4

6 8 6 9

8 ; 9 8

1 0 1 1

(k) (i) (h) (a)

9

0 ;

1 8

6

Xdj. Frdm tc` ubfdjaà fngur` df anh , w` fnba tcktbumi`r dppdsn t` td 9 ns 1, fdr 0 nt ns 8 kba fdr

; nt ns 6. Frdm tcns r`sujt w` hkb a`fnbnt`jy sky

tckt fngur` (i), (h) kba (a) hkb bdt i` tc` kbsw`r

fngur` ks bumi`r jynbg db tc` dppdsnt` pknr df

surfkh` kr` pr`s`bt db tc` kaekh`bt surfkh`s.

C`bh` fng (k) ns dur kbsw`r.

7/21/2019 5-Cubes & Dice

3/18

M@BQKJ KINJNQ^

@x.0 K an` ns tcrdwb fdur tnm s kba nts fdur anff`r`bt

pdsntndbs kr` gnv`b i`jdw. Fnba tc` bumi`r db

tc` fkh` dppdsnt` tc` fkh` scdwnbg 9.

(n) (nn) (nnn) (nv)

(k) 8 (i) 1 (h) 0 (a) 6

Xdj. C`r`, tc` bumi`r 9 kpp`krs nb tcr`` anh`, bkm`jy

(n), (nn) kba (nv). Nb tc`s` anh`, w` dvs`rv` tckt tc`

bumi`rs 9, 1, ; kba 6 kpp`kr kaekh`bt td 8. Xd,

bdb` df tc`s` bumi`rs hkb i` pr`s`bt dppdsnt`

9. Qc` dbjy bumi`r j`ft ns 0.

C`bh`, 0 ns pr`s`bt db tc` fkh` dppdsnt` 9.

@x.6 Xcdwb i`jdw kr`, fdur anff r`bt pdsntndbs df tc`

skm` anh`. Fnba tc` bumi`r db tc` fkh` dppdsnt`

tc` fkh` scdwnbg 6.

(n) (nn) (nnn) (nv)

(k) ; (i) 9 (h) 1 (a) 0

Xdj. Nb tcns hks , tc` bumi`r 6 kpp`krs nb dbjy twd

anh` frdm wcnhc w` dis`rv` tckt tc` bumi`rs ;,

8 kba 0 kpp`kr kaekh`bt td 6, sd tckt 9 dr 1 hkb

kpp`kr dppdsnt` 6.

Xd, w` i`ngb fnbanbg k bumi`r wcnhc kpp`krs kt

j`kst nb tcr`` df tc` gnv`b anh`. 8 ns suhc k

bumi`r, wcnhc kpp`krs nb (n), (nn) kba (nnn). W`

dis`rv` nb tc`s` anh` tckt, tc` bumi`rs ;, 1, 0kba 6 kpp`kr kae`h`bt td 8. Xd, tc`y hkbbdt

kpp`kr dppdsnt` 8. Qc` dbjy bumi`r tckt hkb

kpp`kr dppdsnt` 8 ns 9.

Xd, 9 hkbbdt kpp`kr dppdsnt` 6.

C`bh`, 1 kpp`krs dppdsnt` 6, sd tckt (h) ns tc`

kbsw`r.

W` kr` bdw nb k pdsntndb td sdjv` tc` fdjjdwnbg

`x`rhns`.

@SKM_J@X

@x.7. Wckt ns tc` bumi`r df strkngct jnb`s nb tc`fdjjdwnbg fngur` 4

(K) ;; (I) ;1 (H) ;6 (A) ;7

Xdj. Qc` fngur` ns jki`jj`a ks scdwb.

K I H A

L E N C G

J F

M B

_ [

YD

@

X

Hj`krjy, tc`r` kr` 8 cdrnzdbtkj jnb`s bkm`jy K@,

JF kba LG.

Qc`r` kr` 0 v`rtnhkj jnb s 5 KL, IE, HN, AC kba @ G.

Qc r kr 6 sjkbtnbg jnb s 5 JH, L@, NF, JN, KG kba HF.

Qcus, tc`r` kr` 8 + 0 + 6 = ;1 strkngct jnb`s nb

tc` fngur`.

C`bh`, tc` kbsw`r ns (i).

@x.3 Hdubt tc` bumi`r df trnkbgj`s nb tc` fdjjdwnbgfngur`.

(K) 3 (I) ;? (H) ;9 (A);1

Xdj. W` fnrst jki`j tc` `btnr` fngur` ks scdwb. Hdubt

tc` bumi`r df snmpj`st trnkbgj s. Qc`s` kr` KIC,

ICD, IHA, IDA, A@F, AFD, FGC kba FCD.

Qcus,tc`r` kr` 3 suhc trnkbgj`s. B`xt hdubt tc`bumi`r df trnkbgj`s wcnhc kr` hdmpds`a df twd

hdmpdb`bts `khc. Xuhc trnkbgj`s kr` CIA, IAF,

AFC kba FCI. Qcus, tc`r` kr` 1 suhc trnkbgj`s.

Qc` tdtkj bumi`r df trnkbgj s nb tc` gnv`b

fngur` = 3 + 1 = ;9

Qcus, (H) ns tc` kbsw`r.

@x.> Cdw mkby squkr`s ad`s tc` fngur` ckv` 4

(K) 6 (I) 7 (H) > (A) ;?

7/21/2019 5-Cubes & Dice

4/18

M@BQKJ KINJNQ^

Xdj. Qc` fngur mky i` jki jjà ks scdwb 5

Qc` squkr`s hdmpdsà df twd hdmpdb`bts `khc,kr` KILE, IHJL, HA@J, J@FG, LJGC kba ELCN.

Qcus, tc`r` kr` 6 suhc squkr`s.

Dbjy db` squkr , LH@G ns hdmpds a df fdur

hdmpdb`bts.

Qwd squkr`s bkm`jy, KHGN kba IAFC kr`

hdmpdsà df `ngct hdmpdb`bts `khc.

Qcus, tc`r` kr` 9 suhc squkr`s.

Qc`r` kr` 6 + ; + 9 = > squkr`s nb tc` gnv`b

fngur`. C`bh`, (H) ns tc` kbsw`r.

@x.;? Cdw mkby pkrkjj`jdgrkms kr` tc`r` nb tc` fngurì`jdw 4

(K) ;1 (I) ;0 (H) ;6 (A) ;3

Xdj. W` hkb jki j tc` fngur` ks scdwb.

Qc` snmpj`st || gms kr` KIF@, IHGF, HACG, @FEN,

FGLE kba GCLJ. Qc`s` kr` 6 nb bumi`r.

Qc` || gms hdmpdsà df twd hdmpdb`bts `khc,

kr` KHG@, IACF, @GLN, FCJE, KIEN, IHLE kba

HAJL.

Qcus, tc`r` kr` 7 suhc || gms.Qc` || gms hdmpdsà df fdur hdmpdb`bts `khc,

kr` KHLN kba IAJE. n. `. 9 nb bumi`r. Qc`r` ns

dbjy db` || gm hdmpdsà df snx hdmpdb`bts,

bkm`jy, KAJN.

Qcus, tc`r` kr` 6 + 7 + 9 + ; = ;6 pkrkjj`jdgrkms

nb tc` fngur`.

C`bh`, (H) ns tc` kbsw`r.

@x.;; Wckt ns tc` bumi`r df r`htkbgj s nb tc fdjjdwnbg

fngur` 4

(K) 6 (I) 7 (H) 3 (A) >

Xdj. Qc` fngur ns jki jjà ks scdwb 5

Xnmpj`st r`htkbgj`s kr` K@CG, @FNC, FILE, ELHJ

kba GNJA. n. `. tc`r` kr` 0 suhc r`htkbgj`s.

Qc` r`htkbgj`s hdmpdsà df twd hdpdb`bts `khc

kr` KFNG kba FIHJ.

Qcus, tc`r` kr` 9 suhc trnkbgj`s.Dbjy db` r`htkbgj`, bkm`jy KFJA ns hdmpdsà

df 8 hdmpdb`bts kba dbjy db` r`htkbgj`, bkm`jy

KIHA ns hdmpdsà df 0 hdmpdb`bts.

Qcus, tc`r` kr` 0 + 9 + ; + ; = > r`htkbgj`s nb

tc` fngur`.

C`bh`, (a) ns tc` kbsw`r.

@x.;9 A`t`rmnb` tc` bumi`r df p`btkgdbs nb tc`

fdjjdwnbg fngur`.

(K) 0 (I) 6

(H) 3 (A) ;?Xdj. Qc` fngur` mky i` jki jjà ks scdwb nb tcns hks .

X n x _ ` b t kg d bs c k v` i ` `b fd rm à i y t c `

hdminbktndb df tcr`` trnkbgj`s kba twd rcdm

ius`s - KAFCE, HFCEJ, @CEJI, GEJIA, NJIAF

kba LIAFC.

Fdur dtc`r p`btkgdbs kr` fdrmà iy tc` hdmin

bktndb df tcr`` trnkbgj`s kba db` rcdmius -

JHFCM, JI@CM, ILFCM kba IJNFM.

Qcus, tc`r` kr` ;? p`btkgdbs nb tc` fngur`.

C`bh`, (A) ns tc` kbsw`r.

7/21/2019 5-Cubes & Dice

5/18

M@BQKJ KINJNQ^

Anr`htndbs (;-3) 5

Nb `khc df tc` fdjjdwnbg qu`stndbs fdur pdsntndbs df tc`

skm` anh` ckv` i``b scdwb. ^du ckv` td s`` tc`s` fngur`s

kba s`j`ht tc` bumi`r dppdsnt` td tc` bumi`r ks ksl`a

nb `khc qu`stndb.

[.; 0 6 8 ;

8 8 9 1

6 ; ; 6

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns db tc` dppdsnt` surfkh` df

bumi`r 8 4

(K) 1 (I) 6 (H) 0 (A) ;

[.9 1 ; 0 0

9 0 ; 6

0 9 6 1

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns dppdsnt` td bumi`r ; 4

(K) 1 (I) 6 (H) 9 (A) 8

[.8 0 6 9 0

8 9 0 6

1 ; 1 8

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns dppdsnt` td bumi`r 0 4

(K) 6 (I) 0 (H) ; (A) 8

[.1 9 9 8 ;

6 1 ; 6

1 8 9 0

(n) (nn) (nnn) (nv)


(K) 1 (I) 6 (H) 0 (A) 8

[.0 ; 1 8 1

9 6 0 ;8 ; 9 9

(n) (nn) (nnn) (nv)


(K) 1 (I) 6 (H) ; (A) 8

[.6 6 9 0 8

8 0 8 9

; 1 ; 6

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns db tc` dppdsnt` surfkh` df

bumi`r 8 4

(K) 9 (I) 8 (H) 1 (A) 6

[.7 K A H H

@ H A I

@ I K

(n) (nn) (nnn) (nv)

H

Wcnhc j`tt`r wnjj i` dppdsnt` td j`tt`r A 4(K) K (I) I (H) @ (A) F

[.3 =

D

=

(n) (nn) (nnn) (nv)

=

Wcnhc symidj wnjj hdm` dppdsnt` td symidj

(K) ? (I) = (H) (A)

Anr`htndb (>-98) 5

Nb `khc df tc` fdjjdwnbg qu`stndbs, s`j`ht tc` hdrr`ht dptndb

fdr tc` qu`stndb ksl a.

[.> ;

0

(n)

96

1

9

(nn)

(n) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 9 4

(K) 0 (I) ; (H) 6 (A) 8

(nn) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 6 4

(K) ; (I) 0 (H) 1 (A) 8

(nnn) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 14

(K) 8 (I) 0 (H) ; (A) 9

[.;? 1

;

(n)

99

6

8

(nn)

Wcnhc wnjj i` bumi`r kt tc` idttdm nf 0 ns kt

tdp 4

(K) ; (I) 9 (H) 8 (A) 6

@S@YHNX@

7/21/2019 5-Cubes & Dice

6/18

M@BQKJ KINJNQ^

[.;; Kjj tc` surfkh`s df tc` anh` hdbtknb anff r`bt

bumi`rs nb tc` fdrm df adts. Hdbsna`r idtc tc`

fngur`s df anh` kba t`jj cdw mkby adts kr`

hdbtknb`a db tc` fkh` dppdsnt` td tc` fkh` tckt

hdbtknbs fdur adts 4

(K) 9 (I) 8 (H) 0 (A) 6

[.;9 Wckt ns tc` bumi`r df adts db tc` fkh` dppdsnt`

td tc` fkh` tckt hdbtknbs 9 adts 4

(K) ; (I) 8 (H) 1 (A) 6

[.;8 Db tc` iksns df twd fngur`s df anh , ydu ckv td

t`jj wckt bumi`r wnjj i` db tc` dppdsnt` fkh` df

bumi`r 0 4

8

6

(n)

1;

0

8

(nn)

(K) ; (I) 8 (H) 1 (A) 0

[.;1 Wc`b bumi`r ; ns kt tc` tdp, wcnhc bumi`r wnjj

i` kt tc` idttdm 4

(K) ; (I) 9 (H) 8 (A) 6

[.;0 Wcnhc kjpcki`t wnjj hdm` db tc` dppdsnt` surfkh`

db wcnhc kjpcki`t F kpp`krs 4

K A F

H @ @

H

(n) (nn) (nnn)

I K

(K) I (I) K (H) H (A) @

[.;6 Wcnhc symidj wnjj kpp`kr db tc` dppdsnt`

surfkh` td tc` symidj 'S' 4

(n)

(K) (I) = (H) (A) D

[.;7 Qwd pdsntndb df k anh` kr` scdwb i jdw. Wc`b

tc`r` kr` twd adts kt tc` idttdb, tc` bumi`r df

adts kt tc` tdp wnjj i`

(K) 9 (I) 8 (H) 0 (A) 6

[.;3 Qcr`` pdsntndb df tc` skm` anh` kr` gnv`b i`jdw.

Dis`rv` tc` fngur`s hkr`fujjy kba t`jj wcnhc

bumi`r wnjj hdm` nb pjkh` df '4'

6 0 9

; 8 1

1

(n) (nn) (nnn)

8 4

(K) ; (I) 6 (H) 8 (A) 0

[.;> Db tc` iksns df tc` fdjjdwnbg fngur`s ydu ckv`

td t`jj wcnhc bumi`r wnjj hdm` nb pjkh` df '4'

(K) 9 (I) 8 (H) 6 (A) 1

[.9? Wcnhc j tt`r wnjj hdm` db tc` ijkbl surfkh`4]

I K K

K @ 4

F

(n) (nn) (nnn)

A I

(K) H (I) K (H) A (A) @

[.9; Wcnhc symidj wnjj kpp`kr nb pjkh` df '4'

= D

4

D

(n) (nn) (nnn)

(K) (I) = (H) (A)

7/21/2019 5-Cubes & Dice

7/18

M@BQKJ KINJNQ^

[.99 Wcnhc bumi`r wnjj hdm` nb pjkh` df '4'

9 8 1

; 1 8

0

(n) (nn) (nnn)

8 4

(K) ; (I) 6 (H) 0 (A) 9

[.98 K hui` db wcds` fkh s j`tt`rs ckv` i``b wrntt`b,

ns scdwb i`jdw nb anff`r`bt pdsntndbs ks hkb i`

s``b frdm anff`r`bt anr`htndbs. Fnba tc` mnssnbg

j`tt`rs 4

M M P P

A A ^ W

W 4

(n) (nn) (nnn) (nv)

X A

(K) X (I) A (H) ^ (A) W

Anr`htndbs (91-9>) 5

Qc` fngur` (k) gnv`b i`jdw ns tc` ubfdjaà pdsntndb df k

huinhkj anh`. Nb `khc df tc` fdjjdwnbg qu`stndbs tcns

ubfdjaà fngur` ns fdjjdwà iy fdur anff`r`bt fngur`s df

anh`. ^du ckv` td s`j`ht tc` fngur` wcnhc ns na`btnhkj td

tc` fngur` (K).

[.91

;

6 9

8

1 0

(k)

9 0 8 0

6 6 9 6

9 ;

(K) (I) (H) (A)

1 6

[.90

;

1 8 6

9

0

(k)

8 6 9 ;

9 1 8 1

8 8

(K) (I) (H) (A)

0 0

[.96K

H

A I

@

F

(k)

@ H @ F

H K H I

I K

(K) (I) (H) (A)

A K

[.97

=

+

(k)

+ =

(K) (I) (H) (A)

[.93

F

K @

I

H A

(k)

I @ I @

F F @ A

A K

(n) (nn) (nnn) (nv)

@ H

(K) (n) dbjy (I) (nn) dbjy

(H) (n) kba (nnn) dbjy (A) Kjj

7/21/2019 5-Cubes & Dice

8/18

M@BQKJ KINJNQ^

[.9>

D

(K) (n) dbjy (I) (nn) dbjy

(H) (n) kba (nnn) dbjy (A) (nv) dbjy

Anr`htndb 5

Nb qu`stndbs 8? td 80, fdur anh`s kr` scdwb. Qc` fnrst

tcr`` anspjky kjpcki`ts nb sdm` s`qu`bh`s. Fnba tc`

s`qu`bhnkj ruj` kba fnjj tc` ijkbl spkh`s nb tc` fdurtc anh`

frdm tc` gnv`b fdur kjt`rbktnv`s.

X`qu`bh` df j`tt`rs

K I H A @ F G C N E L J M

B D _ [ Y X Q P V W S ^ U

[.8?

Q

V

S

E

M

_

X

K

@

N

(K) J, B (I) H, F(H) @, D (A) M, G

[.8;

S

K

P

J

Q F

@

C

I

_

(K) M, E (I) X, P

(H) X, V (A) J, G

[.89

_

B

L

D

U

W

Y

[

P E

(K) [, X (I) Q, @

(H) Q, F (A) B, U

[.88

N

M

@

J

@ K

^

Y

J C

(K) Q, _ (I) S, U

(H) X, V (A) A, E

[.81

H

K

F

H

C B

@

C J

M

(K) [, J (I) [, V

(H) _, X (A) C, [

[.80

Y

Q

V

I

V C

F

D

L X

(K) H, N (I) B,

(H) F, M (A) J, U

[.86 Cdw mkby strkngct jnb`s kr` tc`r` nb tc` fdjjdwnbg

fngur` 4

(K) > (I) ;; (H) ;0 (A) 13

[.87 Wckt ns tc` bumi`r df strkngct jnb`s nb tc`

fdjjdwnbg fngur` 4

(K) ;? (I) ;9 (H) ;8 (A) ;7

[.83 Wckt ns tc` bumi`r df strkngct jnb`s nb tc`

fdjjdwnbg fngur` 4

(K) ;8 (I) ;0 (H) ;7 (A) ;>

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M@BQKJ KINJNQ^

[.8> Xtkt` tc` mnbnmum bumi`r df strkngct jnb`s

r`qunr`a td mkl` tc` fngur` gnv`b i`jdw 4

(K) ;6 (I) ;7 (H) ;3 (A) ;>

[.1? Wckt ns tc` bumi`r df strkngct jnb`s kba tc`

bumi`r df trnkbgj`s nb tc` gnv`b fngur` 4

(K) ;? strkngct jnb`s kba 81 trnkbgj`s

(I) > strkngct jnb`s kba 81 trnkbgj`s

(H) > strkngct jnb`s kba 86 trnkbgj`s

(A) ;? strkngct jnb`s kba 86 trnkbgj`s

[.1; Cdw mkby trnkbgj`s kba squkr`s kr` tc`r` nb tc`

fdjjdwnbg fngur` 4

(K) 93 trnkbgj`s, 0 squkr`s

(I) 91 trnkbgj`s, 1 sqkur`s

(H) 93 trnkbgj`s, 1 squkr`s

(A) 91 trnkbgj`s, 0 squkr`s]

[.19 Hdubt tc` bumi`r df trnkbgj`s kba squkr s nb tc`

fdjjdwnbg fngur` 4

(K) 93 trnkbgj`s, ;? squkr`s

(I) 93 trnkbgj`s, 3 squkr`s

(H) 89 trnkbgj`s, ;? squkr`s

(A) 89 trnkbgj`s, 3 squkr`s

[.18 Hdubt tc` bumi r df trnkbgj s nb tc` fdjjdwnbg

fngur` 4

(K) ;0 (I) ;6 (H) ;7 (A) ;3

[.11 Cdw mkby r`htkbgj`s ad`s tc` fdjjdwnbg fngur`

ckv` 4

(K) ;? (I) ;9 (H) ;8 (A) ;1

[.10 Cdw mkby trnkbgj s kr` tc r` nb tc fngur` i`jdw 4

(K) 0 (I) 6 (H) 3 (A) ;?

[.16 Hdubt tc` bumi`r df trnkbgj`s nb tc` fngur` gnv`b

i`jdw 4

(K) ;3 (I) 9? (H) 91 (A) 97

[.17 Hdubt tc` bumi`r df trnkbgj`s nb tc` fdjjdwnbg

fngur` 4

(K) 97 (I) 90 (H) 98 (A) 9;

[.13 Hdubt tc` bumi`r df squkr`s nb tc` fngur` gnv`b

i`jdw 4

(K) ;8 (I) ;6 (H) ;> (A) 9?

[.1> Cdw mkby r`htkbgj`s kr` tc r` nb tc` gnv`b

fngur`4

(K) 6 (I) 7 (H) 3 (A) >

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M@BQKJ KINJNQ^

[.0? Cdw mkby trnkbgj`s ad`s tc` fdjjdwnbg fngur`

ckv 4

(K) 86 (I) 1? (H) 11 (A) 13

[.0; Cdw mkby squkr`s kr` tc`r` nb tc` fngur` gnv`b

i`jdw 4

(K) ;9 (I) ;8 (H) ;6 (A) ;7


fngur` 4

(K) 3 (I) ;? (H) ;; (A) ;9

[.08 Cdw mkby trnkbgj s kr` tc`r` nb tc` fdjjdwnbg

fngur` 4

(K) 9? (I) 91 (H) 93 (A) 89

[.01 Cdw mkby trnkbgj s kr` tc`r` nb tc` fdjjdwnbg

fngur` 4

(K) ;9 (I) ;8 (H) ;1 (A) ;0


fngur` 4

(K) 98 (I) 97 (H) 9> (A) 8;

[.066 6

8 9 1 6 1

0 ;

1 99

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns db tc` dppdsnt` 6 4

(K) ; (I) 9 (H) 8 (A) 1

[.07 ; 88

8 81

1

90

0

;

9

(n) (nn) (nnn) (nv)

Wckt bumi`r ns dppdsnt` 1 4

(K) ; (I) 9 (H) 0 (A) 6

[.03

(n) (nn) (nnn) (nv)

Wckt bumi`r ns dppdsnt` 8 4

(K) 9 (I) 8 (H) 1 (A) 6

[.0>

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns db tc` fkh` dppdsnt` 1 4

(K) ; (I) 9 (H) 8 (A) 0

[.6?

(n) (nn) (nnn) (nv)

Wcnhc bumi`r ns dppdsnt` 8 4

(K) ; (I) 9 (H) 1 (A) 6

[.6; Nf tc` tdtkj bumi`r df adts db dppdsnt` fkh`s df

k huinhkj ijdhl ns kjwkys 7, fnba tc` fngur` wcnhc

ns hdrr`ht 4

(K) (I) (H) (A)

Anr`htndb 5

[u`stndb 69 td 61 kr` iks`a db tc` fdjjdwnbg njjustrktndbs,

Wcnhc kr` fdur vn`ws df k hui`.

(n) (nn) (nnn) (nv)

[.69 Qc` symidj kt tc` idttdm df (nv) ns -

(K) (I) (H) (A)

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M@BQKJ KINJNQ^

[.68 Qc` symidj dppdsnt` tc` fkh` ckvnbg tc symidj =

ns -

(K) (I) (H) (A)

[.61 Qc` symidj dppdsnt` tc` fkh` ckvnbg trnkbgj` ns-

(K) (I) (H) = (A)

[.60 Nf tc` tdtkj bumi`r df adts db dppdsnt fkh`s df

k huinhkj ijdhl ns kjwkys 7, fnba tc` fngur` wcnhc

ns hdrr`ht.

(K) (I) (H) (A)

[.66

(n) (nn) (nnn)

Wckt scduja i` tc` bumi`r dppdsnt` 8 4(K) ; (I) 6 (H) 0 (A) 1

[.67

(n) (nn) (nnn)

Cdw mkby adts jn` dppdsnt` 9 adts 4

(K) ; (I) 8 (H) 0 (A) 6

[.63

(n) (nn) (nnn)Fnba tc` bumi`r df adts db tc` fkh` dppdsnt` tc`

fkh` i`krnbg 8 adts 4

(K) 0 (I) 6

(H) 1 (A) Hkbbdt i` a`t`rmnb`a

[.6>

(n) (nn) (nnn)

Wckt bumi`r dhhur kt tc` idttdm fkh` nb tc`

tcr`` pdsntndbs df tc` skm` an` 4

(K) 6, 6, 9 (I) 0, 6, ;

(H) 0, 0, 0 (A) 6, 0, 9

[.7?

(n) (nn) (nnn)

Fnba tc` kjpcki`t dppdsnt` K.

(K) I (I) H (H) A (A) @

[.7;

(n) (nn) (nnn)

K hui` cks snx anff`r`bt symidjs arkwb dv`r nts

snx fkh`s. Qc` symidjs kr` adt, hnrhj`, trnkbgj`,squkr`, hrdss kba krrdw. Qcr`` anff`r`bt pdsntndbs

df tc` hui` kr` scdwb nb fngur`s (n), (nn) & (nnn).

(n) Wcnhc symidj ns dppdsnt` tc` adt 4

(K) Hnrhj` (I) Qrnkbgj`

(H) Krrdw (A) Hrdss

(nn) Wcnhc symidj ns dppdsnt` tc` Krrdw 4

(K) Hnrhj` (I) Qrnkbgj`

(H) Adt (A) Hrdss

(nnn) Wcnhc symidj dhhurs kt tc` idttdm df fng. (nn).

(K) Krrdw (I) Qrnkbgj`

(H) Hnrhj` (A) Adt

[.79

(S) (^) (U)

(n) Wcnhc bumi`r jn`s kt tc` idttdm fkh` df tc`

an` S 4

(K) ; (I) 9 (H) 8 (A) 1

(nn) Wcnhc bumi`r jn`s kt tc` idttdm fkh` df tc`

an` ^ 4

(K) 6 (I) 0 (H) 9 (A) ;

(nnn) Wcnhc bumi`r jn`s dppdsnt` 6 4

(K) ; (I) 9 (H) 1 (A) 0

(nv) Wcnhc bumi`rs kr` cnaa`b i`cnba tc` bumi`rs

6 kba 0 nb tc` an` U 4

(K) ; & 1 (I) ; & 8

(H) 1 & 8 (A) ; & 9

(v) Wcnhc df tc` cnaa`b bumi`rs kaekh`bt td 0 nb

an` S kr` hdmmdb td tc` cnaa`b bumi`rs kaekh`bt

td 0 nb an` U 4

(K) ; & 1 (I) 9

(H) 6 (A) Bdb`

[.78 Qwd pdsntndbs df k anh` kr` scdwb i`jdw. Nf ;ns

kt tc` idttdm, wcnhc bumi`r wnjj i` db tc`

tdp 4

(n) (nn)

(K) 9 (I) 8 (H) 1 (A) 0

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M@BQKJ KINJNQ^

KBXW@Y L@^

86.\I] Qc` fngur` ns jki`jj`a ks scdwb-

Cdrnzdbtkj jnb`s kr` A@, FC, NJ kba IH n. `. 1 nb

bumi`r. Xjkbtnbg jnb`s kr` NM, FB, AD, KH, KI,

@M, kba CB n. `. 7 nb bumi`r.

Qdtkj bumi`r df jnb`s ns 1 + 7 = ;;.

87.\I] W` hkb jki`j tc` fngur` ks scdwb-

[u`s. ; 9 8 1 0 6 7 3 > (n) > (nn) > (nnn) ;? ;; ;9 ;8 ;1 ;0 ;6 ;7 ;3 ;> 9?

Kbs. K K H H A H K K A K I I A K H A H A I K I H

[u`s. 9; 99 98 91 90 96 97 93 9> 8? 8; 89 88 81 80 86 87 83 8> 1? 1; 19Kbs. K K H H A I H I I H K H K I K I I K I H K H

[u`s. 18 11 10 16 17 13 1> 0? 0; 09 08 01 00 06 07 03 0> 6? 6; 69 68 61

Kbs. H A A H H I A A A A A A H K K H K H H A K K

[u`s. 60 66 67 63 6> 7? 7;(n) 7;(nn) 7;(nnn) 79(n) 79(nn) 79(nnn) 79(nv) 79(v) 78 71 70

Kbs. K I H I H A A I H H H K I A I A K

CNBQX & XDJPQNDB

Nb tcns fngur` 5

Qc` cdrnzdbtkj jnb`s kr` KH, CA kba G@ n. `. 8 nb

bumi`r 5tc` v`rtnhkj jnb`s kr` KG, IF kba H@ n. `. 8 nb

bumi`r5 kba tc` sjkbtnbg jnb`s kr` K@, HA, KA,

HG, A@ kba GA n. `. 6 nb bumi`r.

Qcus, tc`r` kr 8 + 8 + 6 = ;9 jnb`s nb kjj.

83.\K] Qc` fngur` mky i` jki`jj a ks fdjjdws 5

V rtnhkj jnb`s kr` KA, @C, NJ, FG, IH kba EL n. `. 6.

Cdrnzdbtkj jnb s kr` NE, KI, @F, MB, CG, AH, JL

n. `. 7.

Qdtkj bumi`r df jnb`s ns 6 + 7 = ;8

[.71

(n) (nn)

Wckt ns tc` bumi`r df adts kt tc` idttdm fkh`

df tc` j`ft ckba sna` anh` 4(K) 8 (I) 1 (H) 0 (A) 6

[.70 Qwd pdsntndbs df k anh` wntc ; td 6 adts db nts

sna`s kr` scdwb i`jdw. Nf tc` anh` ns r`stnbg db

tc` sna` wntc tcr`` adts, Wckt wnjj i` tc` bumi`r

df adts db tc` sna` kt tc` tdp 4

(n) (nn)

(K) ;dr 0 (I) 9 (H) 8 (A) 6

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M@BQKJ KINJNQ^

Qc`r` ns dbjy db` trnkbgj`s hdmpdsà df tw`jv`

hdmpdb`bts. Qcns trnkbgj` ns KIH.

C`bh`, tc`r` kr` ;? + ;? + 9 + 7 + 6 + ; = 86

trnkbgj`snb tc` fngur`.

[.1;\K] Qc` fngur` mky i` jki`jjà ks scdwb 5

Qrnkbgj`s

Xnmpj st trnkbgj`s kr` KIN, ING, GNC, CNK, IHE,

HFE, FGE, GIE, HAL, A@L, @FL, kba FHL n. `. ;9

nb bumi`r.

Qrnkbgj`s hdmpdsà df twd hdmpdb`bts kr` KIG,

IGC, GCK, CKI, IHF, HFG, FGI, GIH, HA@,

A@F, @FH kba FHA. n.`. ;9 nb bumi`r.

Qrnkbgj`s hdmpdsà df fdur hdmpdb`bts kr` KGH,

IFA, CIF kba GH@ n. `. 1 nb bumi`r.

Qcus, tc`r` kr` ;9 + ;9 + 1 = 93 trnkbgj`s nb tc`

fngur`.

Xqukr`s

Xqukr`s hdmpdsà df twd hdmpdb`bts kr` INGE

kba HEFL n. `. 9 nb bumi`r.

Xqukr`s hdmpdsà df fdur hdmpdb`bts kr`

KIGC,IHFG kba HA@F n.`. 8 nb bumi`r.

19.\H] W` mky jki`j tc` fngur` ks scdwb 5

Qrnkbgj`s

Qc` snmpj`st trnkbgj s kr` NE[, EL[, LJ[, JM[,

MB[, BD[, D_[ kba _N[ n.`. 3 nb bumi`r.

Qc` trnkbgj`s hdmpdsà df twd hdmpdb`bts kr`

KI[, IH[, HA[, A@[, @F[, FG[, GC[, CK[,

NL[, LM[, MD[ kba DN[ n. `. ;9 nb bumi`r.

8>.\I] Qc` fngur` mky i` jki`jj a ks fdjjdws 5

Cdrnzdbtkj jnb`s kr` NE, KI, CG kba AH n. `. 1 trnkbgj`s nb tc`

fngur`.

06.(K) Frdm Fng.(n), (nn) kba (nnn), nt ns hj`kr tckt tc`

bumi`rs 8, 9, 1 kba 0 jn` kaekh`bt td tc` bumi`r

6. sd, ; jn`s dppdsnt` 6.

07.(K) Frdm fng. (n), (nn) kba (nnn), nt ns hj`kr tckt tc`

bumi`rs ;, 0, 1 kba 9 jn` kaekh`bt tds tc` bumi`r

8. Xd, 6 jn`s dppdsnt` 8. Fdrm fng. (nn) kba (nnn), nt

fdjjdws tckt 8, 0 kba 9 jn` kaekh`bt td 1. Xd, ; kba

6 hkb jn` dppdsnt` 1. Iut 6 jn`s dppdsnt` 8.

Qc`r`fdr`, ; jn`s dppdsnt` 1.

03.(H) Frdm, fng.(n), (nnn) kba (nv), nt ns hj`kr tckt bumi`rs 6,

;, 0 kba 9 jn` kaekh`bt td 8. Xd, 1 jn`s dppdsnt` 8.

0>.(K) Frdm fng. (nnn) kba (nv), nt ns hj`kr tckt tc` bumi`r 0,

6, 9 kba 8 jn` kaekh`bt td 1. Xd ; jn`s dppdsnt` 1.

6?.(H) Frdm fng. (n), (nnn) kba (nv), w` fnba tckt tc` bumi`rs

kaekh`bt td 1 kr` 0, 6, ; kba 9. Xd, 8 jn`s dppdsnt`

1 n.`. 1 jn`s dppdsnt` 8.

6;.(H) Xnbh` tc` sum df tc` bumi`r df adts db dppdsnt`

fkh`s df tc` ijdhl ns kjwkys 7, w` hkbbdt g`t ;

adt kaekh`bt 6 adts, 9 adts kaekh`bt 0 adts dr 8

adts kaekh`bt 1 adts. Xd, tc` fngur` (K), (I) kba

(A) hkbbdt i` hdrr`ht.

69.(A) Frdm fng. (n) (nnn) kba (nv), w` dis`rv tckt tc`

symidjs D D = & jn` kaekh`bt td . sd

jn`s dppdsnt` .

68.(K) Frdm fng. (nn) kba (nnn), nt ns hj`kr tckt tc` symidjs

, D kba jn` kaekh`bt td = sngb.

Xd, D jn`s dppdsnt` =

61.(K) Frdm fng.(nnn) kba (nv), nt fdjjdws tc` symidjs ,

= & D jn` kaekh`bt td . Xd db` df tc` symidjs

Ddr hkb jn` dppdsnt` . Iut jn`s

dppsnt` . Xd, D jn`s dppdsnt` .

60.(K) Xnbh` tc` sum df tc` bumi`r df adts db dppdsnt`

fkh`s df tc` ijdhl ns kjwkys 7, w` hkbbdt g`t ;

adt kaekh`bt 6 adts, 9 adts kaekh`bt 0 adts dr 8

adts kaekh`bt 1 adts. Xd, tc` fngur`s (I), (H) kba

(A) hkbbdt i` hdrr`ht.

66.(I) Frdm tc` tcr ` gnv`b fngur`s, nt ns hj`kr tckt tc`

bumi`rs ;, 9, 0 kba 1 kpp`kr kaekh`bt td 8, sd

bdb` df tc`s` hkb kpp`kr dppdsnt` 8. Qc`r`fdr`,

6 kpp`krs dppdsnt` 8.

67.(H) Frdm fngur s (nn) kba (nnn), nt ns hj`kr tckt db ,

tcr`` fdur kba snx adts hkbbdt kpp`kr dppdsnt`

fnv` adts. Qc`r`fdr`, twd adts kpp`kr dppdsnt`fnv` adts.

63.(I) Frdm fng.(nn) kba (nnn) nt ns hj`kr tckt snx, fdur, db`

kba tcr`` adts hkbbdt kpp`kr dppdsnt` twd adts.

Xd, fnv` adts kpp`kr dppdsnt` twd adts. Frdm fng.

(n) kba (nnn) nt ns hj`kr tckt fdur, twd kba db` adts

hkbbdt kpp`kr dppdsnt` tcr`` adts. Kjsd, snbh`

fnv` adts kpp`kr dppdsnt` twd adts sd tc`y

hkbbdt kpp`kr dppdsnt` tcr`` adts. Qc`r`fdr`,

snx adts kpp`kr dppdsnt` tcr`` adts.

6>.(H) Frdm tcr` fngur`ss nt ns hj`kr tckt tc` bumi`rs

9,8,; kba 6 hkbbdt kpp`kr dppdsnt` 1. Xd,0kpp`krs dppdsnt 1. Xnbh` nb `khc db` df tc`

tcr`` anh`, 1 kpp`krs db tc` tdp. Xd, 0 kpp`krs

kt tc` idttdm fkh` df `khc anh`.

7?.(A) Frdm fng. (nn) kba (nnn) nt ns hj`kr tckt H, A, I kba

F hkbbdt kpp`kr dppdsnt` @. Xd, K kpp`krs

dppdsnt` @. n.`. @ ns tc` kjpcki`t dppdsnt` K.

7;. Frdm fng. (n) kba (nnn), nt ns hj`kr tckt adt, trnkbgj`,

hrdss kba krrdw hkbbdt kpp`kr dppdsnt` tc`

hnrhj`. Xd, tc` squkr` jn`s dppdsnt` tc` hnrhj`.

Frdm fng. (nn) kba (nnn), nt ns hj`kr tckt trnkbgj`,

squkr`, krrrdw kba hnrhj` hkbbdt kpp`kr dppdsnt`tc` hrdss. Xd, tc` adt jn`s dppdsnt` tc` hrdss.

Divndusjy, tc` trnkbgj` jn`s dppdsnt` tc` krrdw.

(n) (A) Ks anshussà `krjn`r, tc` hrdss jn`s dppdsnt tc`

adt.

(nn) (I) Ks anshussà `krjn`r, tc` trnkbgj` jn`s dppdsnt`

tc` krrdw.

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M@BQKJ KINJNQ^

(nnn)(H) Xnbh` tc` squkr` jn s kt tc` tdp df fng. (nn) kba tc`

hnrhj` jn`s dppdsnt` tc` squkr`, sd tc` hnrhj` jn`s

kt tc` idttdm df fng. (nn)

79. Frdm fng. (^) & (U), nt ns hj kr tckt tc bumi`rs 1,

8, 9 kba 0 hkbbdt kpp`kr dppdsnt 6. Xd, ; jn s

dppdsnt` 6. Frdm fng. (S) & (^), nt ns hj`kr tckt tc`bumi`rs 0, ;, 6 kba 8 hkbbdt kpp`kr dppdsnt` 1.Xd,

9 jn`s dppdsnt` 1. Divndusjy, 8 jn`s dppdsnt` 0.

(n) (H) 5 Xnbh` 0 jn`s kt tc` tdp dr anh` (S) kba 8 jn`s

dppdsnt` 0, Xd 8 jn`s kt tc` idttdm fkh` df anh`

(S).

(nn) (H) 5 Xnbh` 1 jn`s kt tc` tdp df anh` (^) kba 9 jn`s

dppdsnt` 1, sd 9 jn`s kt tc` idttdm fkh` df anh`

( ).

(nnn) (K) 5 Ks anshuss`a `krjn`r, ; jn`s dppdsnt` 6.

(nv) (I) 5 Qc` bumi`rs wcnhc kr` cnaa`b i`cnba tc`bumi`rs 6 kba 0 nb anh` (U) kr` tc` bumi`rs

jynbg dppdsnt` 0 kba 6 n.`. 8 kba ;.

(v) (A) 5 Nb fng. (S), tc` cnaa`b bumi`rs kaekh`bt td 0 kr`

tcds` jynbg dppdsnt` ; kba 1. Qc`s` kr` 6 kba 9.

Nb fng. (U), tc` cnaa`b bumi`rs kaekh`bt td 0 kr`

tcds` jynbg dppdsnt` 9 kba 6. Qc`s` kr` 1 kba ;.

Hj`krjy, tc`r` ns bd bumi`r hdmmdb td {6, 9 }

kba {1, ;}.

78.(I) 5 Frdm tc` twd fngur`s nt ns hj`kr tckt fdur, fnv , twd

kba snx adts hkbbdt kpp`kr dppdsnt` tcr``

a d ts. Xd , d b ` a d t k p p` k rs d p pd snt ` t cr` `

adts.Qc`r`fdr`, nf db` adt ns kt tc` idttdm, tc`b

tcr`` adts wnjj jn` db tc` tdp.

71.(A) 5 Frdm tc` twd fngur`s nt ns hj`kr tckt tcr``, fdur,

db` kba fnv` adts hkbbdt kpp`kr dppdsnt` twd

adts. Xd, snx adts kpp`kr dppdsnt` twd adts. Nb

tc`s j`ft ckba sna` anh`, tc`r` kr` twd adts db

tc` tdp< sd, tc`r` scduja i` snx adts kt tc`

idttdm fkh`.

70.(K)5 Frdm tc` twd fngur`s nt ns hj`kr tckt twd, snx kba

fdur adts hkbbdt kpp`kr dppdsnt` tcr`` adts. Xd,

`ntc`r db` dr fnv` adts hkb kpp`kr dppdsnt` tcr``

adts. Qc`r`fdr`, nf tc` anh` ns r`stnbg db tc` sna`

wntc tcr`` adts, tc`b `ntc`r db` dr fnv` adts wnjj

kpp`kr db tc` tdp.

5-Cubes & Dice

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