## 3997 – Big Holes by Elap

Posted by Listen With Others on 19 September 2008

**Friday 29th August, 8.00pm**The now traditional annual puzzle from Elap has arrived at last, eagerly anticipated as ever, at least by me. Once again we can see that there is a theme to unravel, which used to be a comparative rarity with the numerical ones.

I’ll get straight down to business and note that we have factorials this time, another rarity and useful to consider first since they increase in magnitude so rapidly. With no indication to the contrary, I shall assume that we are to work in base 10:

6dn TJ + LP + O! (7)

**14dn Q = (V-OU)! (4)**

7! is the only 4-digit factorial so

**V-OU = 7**and

**Q = 5040**

6dn TJ + LP + O! (7)

11! has 8 digits so

I stall for a while at this stage but then spot 26ac:

**O = 10 (16ac)****3dn W = 3T = 3O + T (2)**

**T =**

**15 (29ac), W = 45, 27dn TTT = 3375**

**32dn U = TO + 2T (3)**

**U = 180, V – OU = 7, V = 1807 (28dn), 32ac H = 18 (21ac ends in 8)**

**23ac P = O + 2U (3)**

**P = 370**

**1dn OOE + 2V = 100E + 3614**

So, 1dn must be of the form

**??14**and**E = 1? (13ac)**The grid now looks like this:

I stall for a while at this stage but then spot 26ac:

**26ac OY + OK + A = 10Y + 10K + A = ?3??1**

**A = ?1 or ??1**

**2dn H + A + TH = 288 + ?1 = ??9**

**E = 19, 1dn = 5514**

**35ac AUU + PQ + U = 32400A + 1864980**

**A = 51, 101, 151, 201 or 251**

**R (36dn) = 3?**

**7ac OO + EU + R = 3520 + R= 355?**

**1ac EYY + O + E + A = 5?4??**

**2dn 288 + A**

There is only one fit for A and Y:

**A = 51, Y (8dn) = 53, 1ac = 53451, 2dn = 339, 35ac = 3517380**

**38ac TO + OK + U + V = 2137 + 10K = ?5?07**

**K = ???7**

**11ac KEY + W + Z = 535????**

Z (31dn) is <1000 so KEY must be in the range 5349036 to 5359843

**K = 5317, 38ac = 55307, R = 33, Z = ?15, 5dn 2E + 2R = 104, 26ac = 53751, 7ac = 3553**

**11ac = 5355079, Z = 815, 37ac AY + U + V + Z = 5505**

I wonder if there is anyone still reading. Gregson stopped reviewing the numerical puzzles once the Listener site began giving the full solutions. I should imagine that most solvers would be interested in the entry point to see if an easier route was possible, as would I be, but after that it is probably not so interesting. Perhaps it is just you reading Elap. Well, I am in your debt for all of the hours of first-rate entertainment that you have provided over the years so shall continue with my solution for a little longer:

**23dn TAW + G = 34425 + G = 351?5**

**G in range 690 to 770 step 10**

**17ac TAW + TG = 4?0??**

**G = 710, 23dn = 35135, 17ac = 45075**

**15ac CO + E + 2C = 12C + 19 = 4???**

**C in range 332 to 415**

**4dn RW + C + 2V = 5099 + C = 55?7**

**C = 408, 4dn = 5507, 15ac = 4915**

**21ac COX + O + H = 4080X + 28**

**Must end with 8**

**7dn AB + G + B = 52B + 710 = 391?8**

**B = 739, 7dn = 39138, 22ac B + 2A + 2R = 907**

**30ac M = THE + V + X = 6937 + X = ???8**

**X ends with 1**

**22dn OX + HR + X = 11X + 594 = 9?7?**

**21ac = 4080X + 28 = ??4???8**

**X = 771, 22dn = 9075, 21ac = 3145708, M (30ac) = 7708**

And the grid is almost finished:

And so it went with the grid complete bar 6dn and 19dn when I call it a day. I have yet to look at sorting the letters and just have the alphabet at the bottom of the page with the found values beneath.

**Saturday, 11.00am**The grid is soon finished with

**L = 400**but I have the feeling that the hardest part is yet to come. I can see that there is no 2 or 6 in the grid or in the numbers assigned to letters – perhaps the sum of the 44 grid entries will have to be decoded and N will appear as 26 or 62. The letters have the following values:

**A = 51, B = 739, C = 408, D = 514, E = 19, F = 515, G = 710, H = 18, I = 543, J = 883, K = 5317, L = 400, M = 7708, O = 10, P = 370, Q = 5040, R = 33, S = 738, T = 15, U = 180, V = 1807, W = 45, X = 771, Y = 53, Z = 815.**

Putting these in order of increasing value reveals:

**OTHER WAY UP LCD FIGS (BXZJVQKM)**

Something of a surprise. Had this method not revealed a message I would have tried sorting the numbers as if alphabetical:

**10, 15, 18, 180, 1807, 19, 33, 370, 400, 408, 45, 5040**etc. But this reads:**OTHUVER PLC W QADFY KIGS BMX Z J**!Up until now I have been using just pencil, paper and a calculator but I use MS Excel to add the 44 grid entries – it is quicker and avoids errors:

**37819173**or upside-down

**ELIGIBLE**to be written below the grid:

All finished at 2.16pm.

**Post Mortem**

**Well, perhaps a more straightforward finish than we have come to expect from Elap but a puzzle that was every bit as enjoyable as his previous offerings. I certainly hope that Elap has as much fun setting them, which is by far the more difficult side of things as I see it.**

Let’s have a closer look at the LCD numbers:

6 looks to make a better G, albeit lower case, when turned upside-down. I wonder if there were technical reasons for using 9 rather than 6 in the puzzle?

The title, BIG HOLES, represents the eight numbers, 53704 918, that were used. An alternative title could have accounted for the remainder:

**Half-a-crown Discount**.

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