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Listener 4216: Evens by Elap

Posted by Dave Hennings on 7 December 2012

Oh dear! This week it’s Elap, he of triangular numbers from two years ago. Here we had a wide open spaced 8×8 grid only populated with clue letters, no bars. Unlike Three-squared, which dealt with triangles, this one had clues to the sides, perimeters and area of squares. Eventually, every row and column would need to be completed thematically, and a couple of lines drawn. So at the bottom of today’s effort is my attempt at a detailed solution.

Listener 4216It was, I have to say, fairly obvious that the two lines which would end up showing what was otherwise missing from the puzzle would be in the shape of the number 7. That was made ever so obvious by the title which was Seven with S moved to the end. However, as with many early penny drops, it didn’t help much.

I started by highlighting those squares which contained the last digit of a clue term, eg the last digit occupied by C3, G3 and V2 in 1ac. Square 6 was the place I started, and two entries went in the grid fairly early on … well, after about 40 minutes! As with many mathematicals, it was like chipping away at a pyramid — although you felt you were progressing, there seemed very little to show for it. But I got there in the end.

Listener 4216 My EntryAfter all the clues were solved, the grid was complete … except for those pesky six empty squares, three in the top left and three in the bottom right. Unlike the triangular number in Elap’s last puzzle, it didn’t take too long to realise and verify that all the rows and columns formed squares if started in the right place. Moreover, the main SE-NW diagonal was a square, not to mention 1ac twice: 81649296 and 92968164.

It has to be said that refining, tarting up and keying in my original notes took well over twice the time that it took me to solve Elap’s puzzle in the first place, which was about 4½ hours. I hope you’ll forgive this first attempt, which is nothing compared to erwinch’s efforts, especially in their lack of colour. I’ll try and do better next time!
 
 
 
 
In this table, I have marked unknown digits with the • character, and even digits with an E, eg 12•E•.

No Clue Details Logic Extra Info
1 Sq 6 A=F2.F2.j2 j2 is a square j2=64
2 Sq 7 S=a2.a2
P=a2.j2
j2=4.a2 a2=16
3 Sq 7 from 2 D4+e2+j5=65536 j5=64•••
D4=1•••
4 Sq 2 S=D2+h4
P=n4+p4
p4=4•••
max(n4+p4)=9999+4999=14998
max(D2+h4)=14999 with 4=3749%
but h4 begins even
h4=2•EE
5 Sq 9 S=t2.t2
P=P3.t2
A=B7+U5+s4
P3=4t2
max(P3)=4.99=396
P3=1•• or 2•• or 3••
but P3=E•6 divisible by 4
P3=216 or 236 or 256 or 276 or 296
t2= 54 or 59 or 64 or 69 or 74
but t2=•E
t2= 54 or 64 or 74
P3=216 or 256 or 296

but max(B7+U5+s4)=9999999+99999+9999
=10109997

max(t2)=56

t2=54
P3=216
u3=416
N4=•216
6 Sq 8 S=e2+g3
P=M2+N4
A=E3+K6+f4

g3=••6
M2=•2
N4=2•EE

M2+N4=4(e2+g3)
M2+N4=•2+22•6
=(12+2206)→(92+2296)
=2218→2388
e2+g3=550*→598*
g3=451→588

(e2+g3)^2=E3+K6+f4
=550^2→598^2
=302500→357604
K6=3•••••
K6 is digit 2 of g3
g3=436 or 536

g3=536
d4=1536
7 Sq 4 S=N2.q3+n3
P=b2.d4
A=d4.k3.r2

d4=1536

(b2.d4/4)^2=d4.k3.r2
b2^2.d4^2=16.d4.k3.r2
b2^2.1536=16.k3.r2
b2^2.96=k3.r2
max(k3.r2)=999.99=98901

digits 2&3 k3 = r2
k3 = 100x+r2

b2=EE
b2=20 with k3.r2=38400 or =22 with k3.r2=46464 or =24 with k3.r2=55296 or =26 with k3.r2=64896 or =28 with k3.r2=75264

trying eg 384.84, 464.64, 466.66, etc
b2=20, k3=480, r2=80 or
b2=24, k3=864, r2=64

b=2•
8 Sq 6 S=T2+t2
P=M2+u3
A=F2.F2.j2

t2=54
M2=•2
u3=416
j2=64

M2+u3=428→508
F2.F2.j2=(428/4)^2→(508/4)^2
=11449 or 12544 or 13689 or 14884 or 16129
but is divisible by j2=64
F2.F2=12544/64=196
F2=14
F2=14
T2=58
M2=32
b2=24

from 6:
e2=26
D4=1214

from 8
k3=864
r2=64

9 Sq 7 S=a2.a2
P=a2.j2
A=D4+e2+j5

a2=16
D4=1214
e2=26

a2.a2=256
D4+e2+j5=65536
=1214+26+j5
j5=64296
10 Sq 9 S=t2.t2
A=B7+U5+s4

B7=••61214
U5=•••58
t2=54

B7+U5+s4=8503056
min(U5)=10058
B7=8461214
11 Sq 5 S=T2+W4+a2
P=i5+j2
A=P8+a8+u7

T2=58
W4=•••4
a2=16
j2=64

T2+W4+a2=58+•••4+16=•••8
i5+j2=4.•••8=•••••8
i5=••••8
i5=25••8
A6=816••2
12 contd i5+j2=25008+64→25998+64
=25072→26062
T2+W4+a2=6268→6515
W4=6194→6441
but W4=•EE4
W4=6EE4
=6024 or 6044 or 6064 or 6084 or 6204 or 6224 or 6244 or 6264 or 6284 or 6404 or 6424
i5+j2=24392 or 24472 or 24552 or 24632 or 25112 or 25192 or 25272 or 25352 or 25432 or 25913 or 25992

but digit 4 i5 = digit 3 W4
i5=25128 with W4=6224 or 25928 with W4=6424
with A=39664804 or 42224004

A=P8+a8+u7=216•8•32+16••1854+416••18
min(A)=21608032+16001854+4160018
=41769904
A=42224004

W4=6424
i5=25928
n4=5928
s4=9284
13 Sq 9 S=t2.t2
A=B7+U5+s4

t2=54
B7=8461214
s4=9284

A=8503056
=8461214+U5+9284
U5=32558
14 Sq 2 S=D2+h4
P=n4+p4
A=S4.h4+S5

n4=5928
p4=4296
S4=2558
S5=25584

P=10220
P=2556=12+h4
h4=2544
15 Sq 4 S=N2.q3+n3
P=b2.d4
P=24.1536=36864
S=9216=22.q3+592
q3=392
q3=392
16 Sq 3 P=m3
S=L3
A=Q4+R4+c4

m3=544
Q4=8•32
R4=•322
c4=4E•2

L3=136

A=18496
Q4=8•32+
R4=•322+
c4=4••2

by inspection, Q4=8532, R4=5322, c4=4642

L3=136
17 Sq 5 P=i5+j2
A=P8+a8+u7

i5=25928
j2=64

A=42224004
P8=216•8532+
a8=16••1854+
u7= 416••18

by inspection, P8=21698532, a8=16361854, u7=4163618

18 Sq 8 S=e2+g3
A=E3+K6+f4

e2=26
g3=536

A=315844
E3= 214+
K6=3136••+
f4= 1•38

by inspection, K6=313692, f4=1938

19 Sq 1 S=C3+G3+V2
P=D2.H2+J4
A=A6+R6+d4
A=(816002+532216+1536)→
(816992+532216+1536)
=1352754→1353744
S=1162
A=1350244
20 contd By inspection!!
G3=496, H2=96, J4=3496, V2=54

 

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