What is the minimum number of different colours required to paint he given figure such that no two adjacent regions have the same colour?

The squares composed of two components each are BJMI, CKMJ, DLMK and AIML i.e. 4 in number.

The squares composed of three components each are EBMA, BFCM, MCGD and AMDH i.e. 4 in number.

The squares composed of four components each are VWBA, XYCB, ZA₁DC and B₁C₁AD i.e. 4 in number.

The squares composed of seven components each are NOJL, PQKI, RSLJ and TUIK i.e. 4 in number.

There is only one square i.e. ABCD composed of eight components.

There is only one square i.e. EFGH composed of twelve components.

Total number of squares in the figure = 4 + 4 + 4 + 4 + 1 + 1 = 18.

The simplest rectangles are ABQP, PQNO, BCDN, NDEM, MEFL, LFJK, FGHR and RHIJ i.e. 8 in number.

The rectangles composed of two components each are ABNO, BCEM, NDFL, MEJK and FGIJ i.e. 5 in number.

The rectangles composed of three components each are ACDO, BCFL, NDJK and LGIK i.e. 4 in number.

There is only one rectangle i.e. BCJK composed of four components.

Total number of rectangles in the figure = 8 + 5 + 4 + 1 = 18.

The simplest squares are BCNM, CDON, PQIJ and QRHI i.e. 4 in number.

The squares composed of two components each are MNTS, NOUT, STQP and TURQ i.e. 4 in number.

The squares composed of five components each are ACTL, CEFT, TFGI and LTIK i.e. 4 in number.

The squares composed of six components each are BDUS and SUHJ i.e. 2 in number.

There is only one square i.e. MORP composed of eight components.

There is only one square i.e. AEGK composed of twenty components.

Total number of squares in the figure = 4 + 4 + 4 + 2+1 + 1 = 16.

Triangles :

The simplest triangles are IJQ, JKQ, KLQ, LMQ, MNQ, NOQ, OPQ and PIQ i.e. 8 in number. The triangles composed of two components each are ABQ, BCQ, CDQ, DEQ, EFQ, FGQ, GHQ, HAQ, IKQ, KMQ, MOQ and OIQ i.e. 12 in number.

The triangles composed of four components each are ACQ, CEQ, EGQ, GAQ, IKM, KMO, MOI and OIK i.e. 8 in number.

The triangles composed of eight components each are ACE, CEG, EGA and GAC i.e. 4 in number.

Total number of triangles in the figure = 8 + 12 + 8 + 4 = 32.

Squares :

The squares composed of two components each are IJQP, JKLQ, QLMN and PQNO i.e. 4 in number.

The squares composed of four components each are ABQH, BCDQ, QDEF and HQFG i.e. 4 in number.

There is only one square i.e. IKMO composed of eight components.

There is only one square i.e. ACEG composed of sixteen components.

Thus, there are 4 + 4 + 1 + 1= 10 squares in the given figure.

The horizontal lines are AE and JF i.e. 2 in number. The vertical lines are AJ, CH and EF i.e. 3 in number.

The slanting lines are AG, BF, JD, IE, AB, DE, JI and FG i.e. 8 in number.

Total number of straight lines needed to construct the figure = 2 + 3 + 8 = 13.

Triangles :

The, simplest triangles are ABI, BGI, GHI, HAI, BCJ, CFJ, FGJ, GBJ, CDK, DEK, EFK and FCK i.e. 12 in number.

The triangles composed of two components each are ABG, BGH, GHA, HAB, BCF, CFG, FGB, GBC, CDE, DEF, EFC and FGD i.e. 12 in number.

The triangles composed of four components each are AGC, BFD, HBF and GCE i.e.4 in number.

Thus, there are 12 + 12 + 4 = 28 triangles in the given figure.

Squares :

The squares composed of two components each are BJGI and CKFJ i.e. 2 in number.

The squares composed of four components each are ABGH, BCFG and CDEF i.e. 3 in number.

Total number of squares in the figure = 2 + 3 = 5.

Triangles:

The simplest triangles are KJN, KJO, CNB, OEF, JIL, JIM, BLA and MFG i.e. 8 in number.

The triangles composed of two components each are CDJ, EDJ, NKO, JLM, JAH and JGH i.e. 6 in number.

The triangles composed of three components each are BKI, FKI, CJA and EJG i.e. 4 in number.

The triangles composed of four components each are CDE and AJG i.e. 2 in number.

The only triangle composed of six components is BKF.

Thus, there are 8 + 6 + 4 + 2 + 1 = 21 triangles in the given figure.

Parallelograms :

The simplest parallelograms are NJLB and JOFM i.e. 2 in number.

The parallelograms composed of two components each are CDKB, DEFK, BIHA and IFGH i.e.4 in number.

The parallelograms composed of three components each are BKJA, KFGJ, CJIB and JEFI i.e.4 in number.

There is only one parallelogram i.e. BFGA composed of four components.

The parallelograms composed of five components each are CDJA, DEGJ, CJHA and JEGH i.e.4 in number.

The only parallelogram composed of six components is CEFB.

The only parallelogram composed of ten components is CEGA.

Thus, there are 2 + 4 + 4 + 1 + 4+ 1 + 1 = 17 parallelograms in the given figure.

(Here note that the squares and rectangles are also counted amongst the parallelograms).

There are 13 circles in the given figure. This is clear from the adjoining figure in which the centres of all the circles in the given figure have been numbered from 1 to 13.

We shall join the centres of all the circles by horizontal and vertical lines and then label the resulting figure as shown.

The simplest squares are ABED, BCFE, DEHG, EFIH, GHKJ and HILK i.e. 6 in number.

The squares composed of four simple squares are ACIG and DFLJ i.e. 2 in number.

Thus, 6 + 2 = 8 squares will be formed.

The simplest squares are ABGF, BCHG, CDIH, DEJI, FGLK, GHML, HINM, IJON, KLQP, LMRQ, MNSR, NOTS, PQVU, QRWV, RSXW and STYX i.e. 16 in number.

The squares composed of four components each are ACMK, BDNL, CEOM, FHRP, GISQ, HJTR, KMWU, LNXV and MOYW i.e. 9 in number.

The squares composed of nine components each are ADSP, BETQ, FIXU and GJYV i.e. 4 in number.

There is one square AEYU composed of sixteen components.

There are 16 + 9 + 4 + 1 = 30 squares in the given figure.