Reasoning
Since no digit is duplicated, neither number can end in 1, otherwise, the last digit of the answer would already have been used.
Neither number can end in 5 because the answer would then end in 5 or 0.
So the first number can only be 62, 63, 64, 67, or 68.
We can now look at what the second number can end with, and we find that …
if the first number was 62, the second number can only end in 4 or 7.
Why …
not 1 as previously explained
not 2 because we've already used that in the 62
not 3 because 62 x *3 would end in 6, which is already in the 62
not 5 as previously explained
not 6 because we've already used that in the 62
not 8 because 62 x *8 would end in 6, which is already in the 62
We can repeat this for the other possible first numbers and find that …
if the first number was 62, the second number can only end in 4 or 7.
if the first number was 63, the second number can only end in 4, 7, or 8.
if the first number was 64, the second number can only end in 2, 3, 7, or 8.
if the first number was 67, the second number can only end in 2, 3, or 4.
if the first number was 68, the second number can only end in 3 or 4.
Let's check these in turn …
If the first number was 62, the only possible values are:
62 x 14 = 868
62 x 34 = 2108
62 x 54 = 3348
62 x 74 = 4588
or
62 x 17 = 1054
62 x 37 = 2294
62 x 57 = 3534
62 x 87 = 5394
Only 4 calculations are required for each option, as we didn't need to check 62 x 84 as we already know that the answer would end in 8, or 62 x 47 as we already know that the answer would end in 4.
All of these answers fail as they don't contain all of the digits.
Similarly, we can look at 63, then 64, etc. For each of the numbers, we have to check the possible endings, but in each case, only 4 calculations are required (as seen in the examples above).
We (quickly?) find 64 x 58 = 3712.
??
Puzzle 6
Using the letters AAEEIIMMPPTT complete this grid. The grid reads the same across as down.
L
D
A
E
Note: this puzzle is not interactive, and the squares cannot be clicked.
I really fear lions and tigers, but the aniMAL I fear most though is the mouse.
?
Puzzle 8
Hidden in these sentences are the numbers 1 to 10 (in words).
A number might appear in more than one sentence, and they might not be in order, but there is only one way to use all of the sentences and find all ten numbers.
The robins love hiding amongst the smooth reeds. It's always worth looking after your friends, even if they've upset you. Even heavyweight boxers like using soft tissues when they have a cold. To avoid the cliff, I veered sharply to the left. The eggs were boxed thirteen instead of a dozen in each baker's delivery box. Having salmon every day for lunch gets a little boring after a while. The attendance at the local football match exceeded last week's by many thousands. We need to waterproof our boots to make sure we don't get wet. Meeting friends after work allows executives to network effectively. The orchestra sounded magnificent with the three virtuosi xylophonists.
Answer Having salmON Every day for lunch gets a little boring after a while.Meeting friends after work allows executives to neTWOrk effectively.The robins love hiding amongst the smooTH REEds.We need to waterprooF OUR boots to make sure we don't get wet.To avoid the clifF, I VEered sharply to the left.The orchestra sounded magnificent with the three virtuoSI Xylophonists.It's always worth looking after your friendS, EVEN if they've upset you.Even heavywEIGHT boxers like using soft tissues when they have a cold.The eggs were boxed thirteen instead of a dozeN IN Each baker's delivery box.The atTENdance at the local football match exceeded last week's by many thousands.
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