# Sunday Times Teaser 2676 – New Diary

*by Grahamn Smithers*

In a 3 by 2 rectangular grid there are 12 junctions at which lines either meet or cross. When defined as any two junctions that are joined by straight lines in the grid, we can find 30 ‘junction pairs’.

For many years, diaries have been published using various size grids and the 1998 diary was special because its grid had exactly 1998 such junction pairs.

For one year in the next decade (starting in 2014), a diary can be published for which the number of such junction pairs is equal to the year of publication.

(a) What was the grid size of the 1998 diary?

(b) what is the year of the next special diary?

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If the number of junctions=(n+m)(n+1)(m+1)/2

Hence for 1998 1998×2=3996=37x36x3=(35+2)(35+1)(2+1) and

for 2016 2016×2=4032=28x24x6=(23+5)(23+1)(5+1)

for 2025 2025×2=4050=25x18x9=(17+8)(17+1)(8+1)

and may be more examples to follw…